Algebra II Final -
Determine whether the function is a rational function. If so, find the domain and identify the horizontal and vertical asymptotes, and any holes in the graph. If the function is not rational, state why not. f(x) = (x - 1)/((x - 7)(x + 3))
Incorrect: rational; x≠1, 7 or -3; asymptotes at y=0, x=1, x=7 and x=-3 Correct: rational; x≠-3,7 asymptotes at y=0, x=-3, and x=7
Use a graph to solve the system of equations. {x + y = -4 {y = 3x - 8
(1, -5)
Rationalize the denominator. 2 ÷ √5
(2√5)/5
Which of the following is an identity?
(cos(x))/(1 + sin(x)) = (1 - sin(x))/(cos(x)) Mathway Verify the Identity
Simplify. (-3x + 8)/(x² - 49) - (-2x + 1)/(x² - 49)
-1/(x+7)
The turnstiles at the entrance to the State Fair kept track of the number of people entering the fairgrounds, for the first seven hours following the openning of the fair. Find a quadratic equation that models the data shown. (Evernote)
0.083x² - 0.197x + 7.62 Graphing Calculator STAT→Edit Put x values into L1 and y values into L2. 2ND→MODE→STAT→CALC →5:QuadReg
Solve the equation for x: 10^x = 18
1.26
How many different ways can 6 people be seated around a circular table?
120 Solution (n-1)! (6-1)! 5! = 5 × 4 × 3 × 2 × 1 = 120
Evaluate x⁴ - 10x² + 24 when x=3
15
Evaluate the expression to the nearest thousandth. If the expression is undefined, write undefined: In 10
2.303
Using the given information, which data set is the correct choice? (Evernote)
20, 12, 27, 14, 16, 13, 12, 13, 26, 27, 21, 28, 20, 22, 20
Solve the equation. Round your answers to the nearest hundredth. 5^x-1 = 25
3
A box contains 2 green, 6 yellow, and 3 purple balls. Find the probability of obtaining a purple ball in a single random draw.
3/11
A spinner is evenly divided into 8 equal areas and numbered from 1 through 8. What is the probability of spinning a number less than 4 in a single spin?
3/8
Find the 6th and 8th entries in the row 12 of Pascal's triangle.
792;792 Solution Use nCk-₁ to love for nth entry ₁₂C₆-₁ = ₁₂C₅ = 792 ₁₂C₈-₁ = ₁₂C₇ = 792
Use factoring and the Zero-Product Property to find the zeros of the quadratic function. f(x) = x² -7x - 8
8, -1
Evaluate the expression. (-3/8)²
9/64
Solve the system of equations by substitution. {4y = 4 {x + 2y + z = 10 {y - 2z = -5
Incorrect: None of the listed answers are correct Correct: x=5 y=1 z=3
Which of the following is equal to sin²(x)/(1 - cos(x))
Incorrect: None of the listed answers are correct Correct: 1 + cos(x) or 1 + cosθ Mathway Rationalize the Denominator Mathway Verify the Identity+
Classify the following system of equations as inconsistent, dependent, or independent: {-3x -7y -2z = -25 {-6x +21y -6z = 57 {-3x +14y -4z = 40
independent, because there is an intersection
Simplify: (2 + 5i)/(5 + 4i)
(30/41) + (17/41)i
Evaluate the expression. (2/5)⁵
32/3125
Using the given stem and leaf plot and and it's median and mode, select the data set on which it was created. (Evernote)
7, 6.9, 8.2, 7.6, 6, 7.2, 8.4, 7.6, 7.3, 7.8, 6.2, 9.5, 8.9, 6.5, 9
Evaluate the radical expression: 8(⁴√(81)²
72
Simplify 3x² - 4x - 5 + 4x² + 8x - 7
7x² + 4x - 12
Write the system of inequalities whose solution is graphed as the shaded region.
Incorrect: y < 3x - 2 y > -2x + 5 Use mathway; Graph the intersection (and)
Find all the zeros of the polynomial function. f(x) = x² - 10x + 34
Incorrect: 5 ± 6i Correct: 5 ± 3i
In May, Victor bought 24 styrofoam balls and decorated them as toy figurines. In June, he sold 17 figurines. In May, Wanda bought 28 styrofoam balls to decorate and in June, she sold 12 figurines. Which matrix represents all of their May purchases and their June sales?
May June Victor [24 -17] Wanda [28 -12]
Which matrix represents the graph of polygon PQRS? (Evernote)
P Q R S [2 -3 -3 2] [6 2 -7 3]
Write a polynomial equation in standard form by using the given information. P is of degree 3; P(0) = 24; zeros: 3, -4, 2
P(x) = x³ - x² - 14x + 24
Which property is shown by the following statement? (13 + 3)+ 8 = 13 + (3 + 8)
The Associative Property of Addition
The band and the cheerleading squad at a local school are ordering supplies. The supplies they need are listed in the table. (Evernote)
[10 13 5] [ 5 ] [10 16 8] [12] [ 2 ] = [216] [258]
Write the system of equations as a matrix equation. Then solve the system, if possible, by using a matrix equation. If not possible, classify the system. {3x + 2y + z = -2 {3x -2y -z = -22 {3x + 2y -z = -10
[3 2 1] [x] [ -2 ] [3 -2 -1] [y] = [-22] [3 2 -1] [z] [-10] ;(-4, 3, 4)
Find the center and the radius of the circle that has a diameter with the given endpoints. Diameter CD, C(-1, -4), D(5, 4)
center: (2, 0) radius: 5
Given the quadrant of θ in standard position and a trigonometric function value of θ, find exact values for the remaining functions. Quadrant II, sinθ = 2/7
cosθ = -3√5/7 tanθ = -2√5/15 cotθ = -3√5/2 secθ = -7√5/15 cscθ = 7/2 Mathway (Trigonometry) Find the Other Trig Values in Quadrant II
csc(2x) + cot(2x) forms an identity with which of the following?
cot x Use Mathway to "Verify the Identity" csc(2x) + cot(2x) = cot(x)
Classify the polynomial by degree and number of terms. Describe the shape of its graph. 3x³
cubic monomial; 'S' shaped with 2 turns
Find the distance between points B(5, 7) and F(1, 2), and the coordinates of the midpoint of BF.
distance = √41; 6.4 midpoint = (3, 9/2) Mathway
Which of the following is a quadratic function?
f(x) = -1 + 9x² + 7x
Write the equation of the line, in slope-intercept form, that contains the point (-1, - 5) and is perpendicular to the line -8x - 4y = 3.
y = -2x -7
Write the slope-intercept form of an equation of the line that passes through the point (3, - 6) and has the slope m = -4.
y = -4x + 6
Write an equation in slope-intercept form for a line that passes through the given pair of points. (-6,5), (-9,0)
y = 5x/3 + 15
Which inequality is best represented by the graph shown: (Evernote)
y ≤ -x² + 5x
The two hyperbolas in the graph are conjugates, meaning they share the same asymptotes. Given the following equation of the hyperbola represented by the thin curve, find the equation of its conjugate, which is represented by the thick curve. (Evernote) x²/9 - y²/25 = 1
y²/25 - x²/9 = 1
Solve for x. x-7=1
x=8
Use the information to write the appropriate variation equation, and find y for the given values. y varies jointly as x and z. y=98/3 when x=2 and z=7. Find y when x=8 and z=3
y = 7/3(x)(z); 56
Use the information to write the appropriate variation equation, and find y for the given values. y varies jointly as x and the inverse of z. y=-99.4 when x=-11 and z=4. Find y when x=2 and z=-8.
y = 9x/z; -9/4
Find the inverse of the function and determine whether the inverse is a function. y = 7x²
y = ±√(x/7), y is not a function.
Which function represents exponential decay?
y(x) = 5(0.44)^x
Solve the equation. Give exact solutions. Then approximate the solution to the nearest hundredth, if necessary. 4x² = 8
±√2; ±1.41
How many distinct committees of 14 people can be formed if the people are drawn from a pool of 23 people? Use factorials to express the answer.
₂₃C₁₄ = 23!/(9! 14!) Solution nCr = n!/(r!(n-r)!) 23!/(14!(23-14)!) 23!/(14!(9)!) 23!/ (9! 14!)
Evaluate the trigonometric expression. cos(Cot⁻¹1)
√2/2 Mathway (Trigonometry) Find the Exact Value
If a principal of $1250 is invested at an annual interest rate of 4% compounded annually, what is the account balance at the end of 6 years?
$1581.648
Rationalize the denominator. (√11) ÷ (9 + √5)
( 9√11 - √55) / 76
Evaluate the radical expression: (-1/7)(³√-64)⁴
-256/7
Solve the equation or inequality. (2x-2)/(x²-36) ≥ 1/(x+6)
-6 ≤ x ≤ -4 or x ≥ 6
If f(x) = 3x + 4 and g(x) = 2 + x, which is the rule of function 3g(x) - 3f(x)?
-6x - 6
Find the reference angle for 525°.
15° Mathway (Trigonometry) Find the Reference Angle
Find the value of x to the nearest hundredth.
17.2
Evaluate 2x² + 3x + 4 when x=2
18
What is the reference angle for 516° ?
24° Mathway (Trigonometry) Find the Reference Angle
Evaluate: ₇C₄
35
Determine the probability that you will roll a number greater than 1 on a number cube.
5/6
Evaluate: (₁₂C₇ × ₁₄C₁₀)/₁₃C₅
616 Solution ₁₂C₇ = 792 ₁₄C₁₀ = 1001 ₁₃C₅ = 1287 (792 × 1001)/1287 = 616
Evaluate 8^log₈ 9
9
Designer Dolls, Inc. found that the number of dolls sold, N, varies directly as their advertising budget, A, and inversely as the price of each doll, P. Designer Dolls, Inc. sold 8400 dolls when $84,000 was spent on advertising and the price of a doll was set at $70. Determine the number of dolls sold when the amount spent on advertising is increased to $98,000.
9800
Given that F = 32°, G = 55°, and f = 10, solve the triangle. If no such triangles exist, write none. Round to the nearest tenth. (Evernote)
E = 93° , g = 15.5, e = 18.8 Solution F = 32°, G = 55° , E = ?° f = 10, g = ?, e = ? 32° + 55° + ∠E = 180° ∠E = 93° Use Law of Sines 10/sin(32) = g/sin(55) g = 10sin(55)/sin(32) g = 15.5 10/sin(32) = e/sin(93) e = 10sin(93)/sin(32) e = 18.8
Find the real zeros of the function. Give approximate values to the nearest hundredth , if necessary. f(x) = x³ - 4x² + 7x - 12
None of the listed answers are correct
The table shows the number of hybrid cottonwood trees planted in tree farms in Oregon since 1987. Find a cubic function to model the data and use it to estimate the number of cottonwoods planted in 1998. (Evernote)
T(x) = 0.2x³ + 0.3x² + 0.2x + 0.1; 304.8 thousand
Which set of characteristics matches the given standard equation and parabola graph? (Evernote)
focus: (-1, -4) vertex: (-1, 2)
Which of the given ordered pairs are NOT solutions to the system of inequalities? {2x + 3y < 1 {x -3y > 2 (3, 1), (-3, -2), (5, -5), (-2, 4), (-4, 5)
(-4, 5), (3, 1), (-2, 4)
Use the Law of Sines to solve for m∠A to the nearest tenth. (Evernote)
29.8 Solution sin B/b = sin C/c sin B/43 = sin(80°)/45 sin B = 43sin(80°)/45 sin B = 0.94103 sin⁻¹(sin B) = ∠B sin⁻¹((43sin¹(80°)/45)) sin⁻¹(0.94103) = 70.2° (rounded to the nearest tenth) ∠B = 70.2° ∠A + 70.2° + 80° = 180° ∠A = 29.8° Use graphing calculator on degree mode or use Mathway and change answer to degrees.
If 2 blocks are randomly taken from a bag containing 10 blue blocks, 10 red blocks, and 7 yellow blocks, what is the probability of drawing a blue block and a red block?
50/351 or 0.1424501 Solution 10/27 × 10/26 = 0.1424501
Write the product as a polynomial in standard form. (5x² - 2x + 5)(x² - 8x - 2)
5x⁴ - 42x³ + 11x² - 36x - 10
The function d = 9 cos 2t describes a simple harmonic motion, where d is the distance (in meters) an object travels in t seconds. What is the maximum displacement of the object from its resting position?
9 m Solution A = displacement d = A cos wt d = 9 cos 2t A = 9 or 9m
Use the given formula to find the first four terms of the arithmetic sequence. t(n) = 12 - 3n
9, 6, 3, 0 Solution t(1) = 12 - 3(1) 9 t(2) = 12 - 3(2) 6 ...
Solve triangle ABC given that a = 11, b = 13, and c = 15.
A = 45.6° , B = 57.6° , C = 76.9° Solution Use Law of Cosines cos A = (b² + c² - a²)/2bc cos A = (13² + 15² - 11²)/(2 × 13 ×15) cos A = 7/10 cos⁻¹(7/10) = 45.6° cos B = (c² + a² - b²)/2ca cos B = (15² + 11² - 13²)/(2 × 15 × 11) cos B = 59/110 cos⁻¹(59/110) = 57.6° cos C = (a² + b² - c²)/2ab cos C = (11² + 13² - 15²)/(2 × 11 × 13) cos C = 5/22 cos⁻¹(5/22) = 76.9°
Find all solutions of cos(x) - √(1 - 3cos²(x)) = 0
Incorrect: None of the listed answers are correct Correct: No Solution
An English class was taken by 28 students. Their final scores were: three As, five Bs, nine Cs, six Ds, and five Fs. Using the given frequency table, find the mean grade point score, using A = 4, for the entire group of 28 students. Give the answer to the nearest hundredth, if necessary. (Evernote)
Incorrect: [none] Correct: 1.82 (Close to a "C")
Suppose the depth of the tide in a certain harbor can be modeled by y = 25 - 4 cos(π/6t), where "y" is the water depth in feet and "t" is the time in hours. Consider a day in which t = 0 represents 12:00 midnight. For that day, when are high tide and low tide and what is the depth of each?
Incorrect: high tide at 12:00 noon and 12:00 midnight, depth 29 ft low tide at 6:00 a.m. and 6:00 p.m., depth 21 ft Correct: high tide at 12:00 noon and 12:00 midnight, depth 21 ft low tide at 6:00 a.m. and 6:00 p.m., depth 29 ft
Find AC - CB (Evernote)
[-13 19] [-12 -44]
Solve the equation or formula for the variable specified. B = 7a²b, for b.
b = B/(7a²)
Which function is the correct match with the the given graph and vertex (Evernote)
f(x) = -3(x + 1)²
Which system of equations can be classified as independent?
none of these
Solve the proportion (x−8)/8 = 4/5
x=14 or 72/5
Which equation best describes the graph below? (Evernote)
y = -2x + 5
NCUse the frequency table below, with the a total of 18 catalog items sold at four different prices, to find the average price per item. (Evernote)
$16.50
If $1500 is invested at an interest rate of 10%, compounded continuously, determine the balance in the account after 2 years. Use the formula A = Pe^rt.
$1832.10
A T-shirt company estimates that the average cost per shirt can be approximated by the function A(x) = (3.75x + 200)/x, where x is the number of T-shirts made. Find the average cost per T-shirt when the company makes 10 shirts.
$23.75
The inflation rate of the U.S. dollar is 3.1 percent. What this means is that every year, prices increase by 3.1 percent. If a pound of meat cost $2.37 nine years ago, what does it cost now?
$3.12 Solution 2.37(1+0.031)⁹
Mr. Reaich, the ticket agent for a small commuter airline, has developed a model to forecast the total monthly revenue from the airline's passenger service. Under this model, the total revenue, R, is equal to $368 times the number of first-class, F, plus $324 times the number of coach passengers, C. Their flight schedule and overall aircraft capacity limit the total number of passengers carried during a single month to a maximum of 30,775. The number of coach class passengers is at least seven times the number of first-class passengers, and logically, the number of airline passengers in either class cannot be negative. Express the model in mathematical form.
$368F + $324C = R F + C ≤ 30,775 C ≥ 7F F ≥ 0; C ≥ 0
Find the final amount of the investment: $3000 at 8% interest compounded quarterly for 4 years.
$4118.36
Write a proportion that models the statement 10 is to 5 as 16 is to 8.
10/5 = 16/8
Evaluate the expression. (3/8)⁻²
64/9
Which equation describes a parabola?
6y² + 11x - 13y = -1
Erosion gradually reduces the size of a small Pacific island that has a current area of just 460 acres. If the island's area decreases at an annual rate of 0.05%. Find the multiplier for the rate of exponential decay.
Incorrect: 0.95 Correct: 0.9995 Solution 100 - 0.05 = 99.95 99.95/100 = 0.9995
Perform the indicated operations on the given matrices. [3 -6 -7] [-2 -1 7] [-3 -2 4] + [ 3 5 -7] [-4 -1 7] [4 -9 -3]
[ 1 -7 0 ] [ 0 3 -3 ] [0 -10 4]
Divide x³ + 5x² + 9x + 9 ÷ x + 3
x² + 2x + 3
Evaluate the expression. 3 - 14 × 6 ÷ 7 + 3
-6
Find the determinant, and tell whether the matrix has an inverse. det [10 -7] [-5 -6 ]
-95; Yes
Use factoring and the Zero-Product Property to solve the quadratic equation 2x² + 5x -3 = 0
1/2, -3 Use Quadratic equation: (-b ± √((b² - 4ac))/2a
Use a graph, synthetic division, substitution, and factoring to solve the equation. x³ - 7x² + 14x - 8 = 0
2, 1, 4
Find the real zeros of the function. Give approximate values to the nearest hundredth , if necessary. f(x) = x⁴ - 9x³ + 10x² + 90x - 200
4, 5, ±3.16
Which is the equation of the line with slope -4 and y-intercept 5?
4x + y -5 = 0 y = -4x + 5
Find: ₇P₄
840
Write the quadratic equation in vertex form. Give the coordinates of the vertex and the equation of the axis of symmetry. y = -6x² - 24x - 31
Incorrect: y = 6(x + 2)² - 7 x = -2; (2,-7) Correct: y = -6(x + 2)² - 7 x = -2; (-2, -7) Mathway: Find the Vertex Form Find the Axis of Symmetry Find the Vertex (Coordinates)
The function d = - 6 cos 3t describes a simple harmonic motion, where d is the distance (in meters) an object travels in t seconds. What is the frequency?
Incorrect: 3π/2 cycles/seconds Correct: 3/2π Solution d = A cos wt d = -6 cos 3t w = 3 f (frequency) = w/2π f = 3/2π
Write the quadratic equation in vertex form. Give the coordinates of the vertex and the equation of the axis of symmetry. y = x² + 14x + 4
Incorrect: None of the listed answers are correct Correct: y = (x+7)² - 45 x = -7; (-7, -45)
Write the expression csc(2x) + cot(2x) in terms of tan x or cot x.
Incorrect: None of the listed answers are correct Correct: cot(x)
Solve the equation or inequality. 4/(x-3) - 2/(x+4) = 0
x=-11
Simplify. ( (x - 2)/(2x²) ) - ( (2x + 1)/(9x) ) + ( (7x)/(12) )
( 21x³ - 8x² + 14x - 36 ) / 36x²
Simplify the rational expression. (Evernote) ( 3/(x² + 7x + 12) + 1/(x² + 11x + 28) ) ÷ ( 2/(x² + 10x + 21) + 1/x² + 13x + 36) )
( 4(x + 9)(x + 6) ) / ( 3(x² + 12x + 31) )
Use elimination to solve the system of equations. {4x + 2y = -12 {3x - 2y = 5
(-1, -4)
Use a graph to solve the system of equations. {x + y = -1 {y = 3x + 19
(-5, 4)
Find the image of (5, -6) after a counterclockwise rotation of 270° about the origin. (Evernote)
(-6, -5)
Convert 279° to radians.
(31/20)π Mathway (Trigonometry) Convert from Degrees to Radians
Solve the system of equations by substitution. {3x + 4y = 32 {3x + y = 17
(4, 5)
For a circle of radius 8 feet, find the arc length of a central angle of 30°.
(4/3)π feet Solution r = 8 and θ = π/6 (change 30° to radians) s = rθ s = 8(π/6) s = (4/3)π or 4π/3
Express 0.7312 as a geometric series, and write its sum as the ratio of two integers.
(Evernote)
The Student Government secretary, and two teacher's aides are ordering supplies. The supplies they need are listed in the table. (Evernote)
(Evernote)
Find the standard equation of a circle with the given radius and center. radius: 4 center: (3, - 5)
(x - 3)² + (y + 5)² = 16 Mathway: Find the Equation (put coordinate points comma r=4)
Find the standard equation for the ellipse, using the given characteristics taken from the graph. (Evernote)
(x+4)²/16 + (y+3)²/9 = 1
Simplify the rational expression. ( (x² + 14x + 49)/8x ) ÷ ( (x+7)/4x )
(x+7)/2
Find the standard equation for the hyperbola with the given characteristics. center: (5, 4) one focus: (-25, 4) one vertex: (23, 4)
(x-5)²/324 - (y-4)²/576 = 1
Write the equation in standard form and classify the conic section it defines. 2x² + 2y² - 28x + 24y + 162 = 0
(x-7)² + (y+6)² = 4; circle
Find the exact value of cos(345°).
(√6 + √2)/4 Mathway (Trigonometry) Find the Exact Value
Use a graph, synthetic division, substitution, and factoring to solve the equation. x³ - x² - 14x + 24 = 0
-4, 2, 3
Solve the equation by completing the square. Give exact solutions. x² - 2x - 24 = 0
-4, 6
Simplify the expression. ((-u)³(-u⁹)⁸)/(u⁵)³
-u⁶⁰ Solution (-u³ × u⁷²)/u¹⁵ -u⁷⁵/u¹⁵ = -u⁶⁰
Find cot (-290°).
0.364 Mathway (Trigonometry) Find the Exact Value
Find tan 382°.
0.404 Mathway (Trigonometry) Find the Exact Value
Whitney starts the engine on her small private airplane. The engine drives a propeller with a radius of 7.5 feet and its centerline 12.5 feet above the ground. At idle, the propeller rotates at a constant speed of approximately 850 revolutions per minute. The height of one propeller tip as a function of time is given by h = 12.5 + 7.5 sin(850t), where h is the height in feet and t is the time in minutes. Find h when t = 2.5
8.2 ft Solution h = 12.5 + 7.5 sin(850t) h = 12.5 + 7.5 sin(850 × 2.5) h = 8.2
Refer to ∆ ABC below to find the indicated value listed. Find the exact value and the value rounded to the nearest ten-thousandth, if necessary. (Evernote)
8/17 or 0.4706 sinθ = opp./hyp.
Jamila spins a spinner with 3 sections of equal area, like the one below, 25 times. It lands on the 1eight times. What is the experimental probability of spinning a 1?
8/25
Since 1993, Daphne Hamilton has owned a franchise of take-out restaurants called The Burger Barn. The number of customers, C, in thousands, that The Burger Barn has served each year can be modeled by the function C(t) = t² + 38t + 600, where t is the number of years from 1993. Using this model, estimate the number of customers served in 1999.
864,000
Find the reduced row-echelon form of the augmented matrix that represents the following system of equations: (Evernote) {-6x + 9y = -5 {-8x - 4y = 0
Incorrect: [1 0| 5/24 ] [0 1|-5/12] Correct: [1 0| -5/24] [0 1| 5/12
Find cos((1/2)A) and sin((1/2)A) if sin(A)=4/5 and Ais a first-quadrant angle. (Evernote)
Incorrect: cos(A/2) = √3/5 sin(A/2) = 2/5 Correct: cos(A/2) = (2√5)/5 sin(A/2) = √5/5 Solution Pathagorean Identities sin A = 4/5 cos²A = 1 - sin²A cos A = 1 - (4/5)² cos A = 3/5 Use Half-Angle Identities sin(A/2) = √((1 - cos A)/2) sin(A/2) = √((1 - 3/5)/2) sin(A/2) = √5/5 cos(A/2) = √((1 + cos A)/2) cos(A/2) = √((1 + 3/5)/2) cos(A/2) = (2√5)/5 A = θ and the cos((1/2)A) is the same as cos(A/2)
Which inequality has (1, 7) as a solution?
Incorrect: -2x - 5y ≥ -32 Solution Substitute -2(1) - 5(7) ≥ -32 -2 - 35 ≥ -32 -37 ≥ -32 False, not a solution of the inequality
Given sinθ = 5/9, where π/2 < θ < π, find the exact values of sin 2θ and cos 2θ.
Incorrect: -31/81, (20√14)/81 Correct: 10/9, -(4√14)/9
Two cards are randomly selected from a standard 52-card deck. What is the probability of getting 2 diamonds or 2 face cards?
Incorrect: 0.108 Correct: 0.106 or 47/442 Solution (13/52 × 12/51) + (12/52 × 11/51) - (3/52 × 2/51)
Use Pascal's triangle to solve for the value of n: 2(n C 3) = n+4 C ₇ (Evernote)
Incorrect: 2 Correct: 4
Use Pascal's triangle to find the number of ways to choose 3 boxes from 7 boxes.
Incorrect: 21 Correct: 35 Solution ₇C₃ = 35
Solve the equation. Give exact solutions. Then approximate the solution to the nearest hundredth, if necessary. 2(x + 4)² - 48 = 0
Incorrect: 4 ± 2√6; 5.8 or -13.9 Correct: -4 ± 2√6; 0.9 or -8.9
Find the two geometric means between 7 and 3584.
Incorrect: 63, 567 Correct: 56, 448 Solution t₁=7 and t₄=3584 Formula t(n) = t₁r^(n - 1) Substitute t(4) = 7r⁴⁻¹ 3584 = 7r³ divide both sides by 7 r³ = 512 Take cube root of both sides r = 8 t(2)=7 × 8 56 t(3)=7 × 8² 448 7, 56, 448, 3584
Use Pascal's Triangle to determine the probability that you will get three green lights in a row of five lights. Assume red and green are equally likely occurrences.
Incorrect: 7/32 Correct: 5/16 (10/32) Solution (n C k)(p∧k)(1 - p)∧(n-k) (₅C₃)(½)³(1 - ½)⁵⁻³ (₅C₃)(½)³(1 - ½)² 10 × 1/8 × 1/4 = 5/16
Solve sin(x) - √(1 - 3sin²(x)) = 0 given that 0° ≤ x ≤ 360°
Incorrect: 90°, 270° Incorrect: None of the listed answers are correct Correct: 30°, 150°, 210°, 330° Solution sin(x) = √(1 - 3sin²(x)) sin²(x) = 1 - 3sin²(x) sin²(x) + 3sin²(x) - 1 4sin²(x) - 1 (2sin(x) + 1)(2sin(x) - 1) sin(x) = -1/2 and 1/2 x = -30°, 30°, 150°, 210° Add 360° to -30° -30° + 360° = 330° x = 30°, 150°, 210°, 330° Or graph y = sin²(x) y = 1 - 3sin²(x) Intersections: 30°, 150°, 210°, 330°
Solve sin(x) - √(1 - 3sin²(x)) = 0 given that 0° ≤ x ≤ 360°.
Incorrect: 90°, 270° Incorrect: None of the listed answers are correct Correct: 30°, 150°, 210°, 330° Solution sin(x) = √(1 - 3sin²(x)) sin²(x) = 1 - 3sin²(x) sin²(x) + 3sin²(x) - 1 4sin²(x) - 1 (2sin(x) + 1)(2sin(x) - 1) sin(x) = -1/2 and 1/2 x = -30°, 30°, 150°, 210° Add 360° to -30° -30° + 360° = 330° x = 30°, 150°, 210°, 330° Or graph y = sin²(x) y = 1 - 3sin²(x) Intersections: 30°, 150°, 210°, 330°
Describe the value of b for the function y(x) = 1/3(1/b)^x to represent exponential growth.
Incorrect: None of the listed answers are correct Correct: "b" has to be greater than 0 and less than 1. (0 < b < 1)
The relative frequency histogram below represents the age in years of the first 100 children to have their portraits taken at the "See What Develops" photography studio. What is the probability that the next child to have portraits taken will be between 1 and 2 years old?
Incorrect: None of the listed answers are correct Correct: 31/100 or 31% (maybe 0.315 or 32%) Solution .11 + .20 + .31 + 30 + .15 + 24 = 1 (or 100%) Age 1= .20 Age 2= .11 .20 + .11 = .31 (or 31%) 31/100 or 31% Probability
Wilfred is checking his bicycle's wheel. The wheel has a radius of 25 cm, and its center is 54 cm above the ground. Wilfred is rotating the wheel at a constant speed of 230° /s. The height of a point on the tire as a function of time is given by h = 54 + 25 sin(230t), where h is the height in centimeters and t is the time in seconds. Find h when t = 5.5 s. Round your answer to the nearest tenth of a centimeter.
Incorrect: None of the listed answers are correct Correct: 51.8 Solution h = 54 + 25 sin(230t) h = 54 + 25 sin(230 × 5.5) h = 51.8
The magnitude of an earthquake is found by the equation M = 2/3 log E/10^11.8 , where M is the magnitude and E is the energy released. Find the magnitude of an earthquake that released 10^24.4 ergs of energy.
Incorrect: None of the listed answers are correct Correct: M = 8.4 Solution M = 2/3 log₁₀ (10^24.4/10^11.8)
The wattage rating of an appliance is given as "W", in watts, and varies jointly as the resistance, "R", in ohms, and as the square of the current, "I", in amperes. If the wattage is 2 watts when the resistance is 200 ohms and the current is 0.1 amperes, find the wattage when the resistance is 150 ohms and the current is 0.4 amperes.
Incorrect: None of the listed answers are correct Correct: W (wattage) = 24
Solve log₇ 8 - log₇ (x + 3) = log₇ 3, for x
Incorrect: None of the listed answers are correct Correct: x=-1/3 Solution log(x) - log(y) = log(x/y) log₇ 8 - log₇ (x+3) = log₇ (8/(x+3)) log₇ (8/(x+3)) = log₇ 3 (Mathway can solve once the left terms are combined)
For the data given, find the equation of the line of best fit. (Evernote)
Incorrect: None of the listed answers are correct Correct: y = 0.63x + 1.90
Use the set of constraints to find the maximum and minimum values of the objective function C = 5x + 11y Constraints: {x + y ≥ 2 {6x - 6y ≤ 12 {8y ≤ 4x + 16
Incorrect: The maximum value of C is 139 at (8, 9) and the minimum value is 10 at (0, 2). Correct: The maximum value of C is 106 at (8,6) and the minimum value is 10 at (2,0).
Make a conjecture about the pattern of the given data. Find the sum of the 7th row. 3 3 + 6 + 3 3 + 6 + 9 + 6 + 3 3 + 6 + . . . + ( 3n − 3) + ( 3n ) + ( 3n − 3) + . . . + 3
Incorrect: The sums of the rows are 3, 12, 27, . . . . The sum of the nth row is n(2n +3). The sum of the 7th row is 119. Correct: The sum of the rows are 3, 12, 27, . . . . The sum of the nth row is 3(n²). The sum of the 7th row is 147.
Use elimination to solve the system of equations. {-16x - 6y = 170 {8x + 3y = -85
Incorrect: no solutions Correct: Infinitely Many Solutions
Solve: (Evernote) det[x+3 2] [3 x-2] =0
Incorrect: x =-4 Correct: x = 0 Mathway: Find the Determinant
There are 8 cars in a parking lot on a very cold day. Suppose the probability of any one of them not starting is 0.14. What is the probability that exactly 3 of the cars will not start?
Incorrect: ≈ 0.003 Correct: ≈ 0.072 Solution (nCk) × (p)^n × (1 - p)^(n - k) (₈C₃) × (0.14)³ × (1 - 0.14)⁵ = 0.072
Graph and classify the system of equations as independent, inconsistent, or dependent. If the system is independent, find the solution from the graph. {3x + 3y = 2 {3x - y = 8
Independent (2, -2)
The depth of snow at seven different mountain lodges is 13 in., 15 in., 21 in., 17 in., 90 in., 13 in., and 19 in. Find the mean, median, and mode. Tell which measure is the most useful for predicting how deep the snow will be at an eighth lodge.
Mean = 26.9 in Median = 17 in Mode = 13 The median is the most useful.
Find the minimum and maximum values, quartiles, range, and interquartile range for the data set. 27, 8, 9, 48, 19, 10, 29, 42, 2, 50, 44
Minimum = 2 Maximum = 50 Q₁=9; Q₂=27; Q₃=44 Range = 48 IQR = 35 Mathway Find the Five Number Summary, range, and Find the Interquartile Range (H-Spread)
Find the minimum and maximum values, quartiles, range, and interquartile range for the data set. 29, 14, 40, 12, 16, 46, 22, 23, 19, 31, 7, 9
Minimum = 7 Maximum = 46 Q₁=13; Q₂=20.5; Q₃=30 Range = 39 IQR = 17 Q₂ = mean
For a circle of radius 4 feet, find the arc length "s" subtended by a central angle of π/30 radians. Round to the nearest hundredth.
None of the listed answers are correct Solution r = 4 and θ = π/30 s = rθ s = 4(π/30) s = 2π/15
A number cube is tossed 10 times with the following results. 5, 6, 1, 5, 5, 2, 6, 1, 2, 2 Find the range and the mean deviation of the data.
None of the listed answers are correct To find mean deviation minus the mean from each data value, add everything up, then divide by the number of data values. Example: 54, 49, 47, 48, 52 mean of data = 50 54 - 50 = | 4 | = 4 49 - 50 = |-1| = 1 47 - 50 = |-3| = 3 48 - 50 = |-2| = 2 52 - 50 = | 2 | = 2 4 + 1 + 3 + 2 + 2 = 12 12 ÷ 5 = 2.4 Mean deviation = 2.4
Point P is located at the intersection of a circle with a radius of r and the terminal side of an angle θ. Find the coordinates of P to the nearest hundredth. θ = 120° , r = 13
P(- 6.5, 11.26) Solution 120° is in Quadrant II Coordinates: (cos θ = x/r, sin θ = y/r) θ = 120° and r = 13 cos(120°) = x/13 -½ = x/13 x = -13/2 or -6.5 sin(120°) = y/13 √3/2 = y/13 y = 13√3/2 or 11.26 P(-6.5, 11.26)
Use the graph of the feasible region for the set of constraints to find the maximum and minimum values of the objective function C = 3x + 5y. Constraints: {x + y ≥ 1 {4x - 2y ≤ 4 {3y ≤ 3x + 3
The maximum value of C is 29 at (3, 4) and the minimum value is 3 at (1, 0) .
Hans pays $205 in advance on his account at the athletic club. Each time he uses the club, $10 is deducted from the account. Write an equation that represents the value remaining in his account after x visits to the club. Find the value remaining in the account after 11 visits.
V = 205 - 10x; $95
Perform the indicated operations on the given matrices. (Evernote) [-8 -8] + [-2 3] [-6 4] + [3 -1] (2×2 matrice + 2×2 matrice)
[-10 -5] [-3 3]
Write the augmented matrix for the system of equations. {x - y = -9 {5x = 0
[1 -1| -9] [5 0| 0] Solution x=1, y=-1 5x=5
Find [1 17 14][11 3] [19 7] [9 5]
[460 192]
Write the augmented matrix for the system of equations. {8x + 2y + 6z = -3 {3x - y + 10z = 3 {-9x -2y -4z = -7
[8 2 6| -3 ] [3 -1 10| 3 ] [-9 -2 -4 |-7 ]
Ivan Bogdanovich plans to decorate stuffed animals to sell at a crafts fair. The decorations cost $44.00 and the stuffed animals cost $5.25 each. a. Write a function expressing the cost, C(x), of the project in terms of the number of stuffed animals decorated, x. b. Determine the cost of decorating 25 stuffed animals. c. How many stuffed animals can be decorated with a budget of $227.75?
a. C(x) = 5.25x + 44.00 b. $175.25 c. 35
Which set of characteristics matches the given parabola graph and standard equation? (Evernote) x = -1/4 y²
axis of symmetry: y=0 focus: (-1, 0) vertex: (0, 0) Mathway: Graph
Use the following information to find the unknown sides and angles. m∠B = 28°; c=18
b = 8.5; a = 15.9; m∠A = 62°; m∠C = 90° Solution We have two angles m∠A + m∠B + m∠C = 180 m∠A + 28° + 90° = 180° m∠A = 62° Use Law of Sines to solve for side "a" or "b' a/sin A = c/sin C a/sin(62°) = 18/sin(90°) a = 18sin(62°)/sin(90°) a = 15.89 → 15.9 b/sin B = a/sin A b/sin(28°) = 15.9/sin(62°) b = 15.9sin(28°)/sin(62°) b = 8.45 → 8.5
Find the center and the radius of the circle that has a diameter with the given endpoints. Diameter CD, with endpoints C(- 1, 5), D(5, 9)
center: (2, 7) radius: √13; 3.61 Mathway: Find the Midpoint
Identify the conic section produced by the intersection of the plane and the cones in the following diagram.
circle
Show that the function is a quadratic function by writing it in the form f(x) = ax² + bx + c and indentifying a, b and c. f(x) = (5x - 4) (5x + 3)
f(x) = 25x² - 5x - 12 a = 25, b =5, c = -12
Given the graph of y = 3(x + 1), what function would be obtained by moving the graph up 4 and moving it 6 to the right?
f(x) = 3(x - 5) + 4
Solve the nonlinear system of equations. {x² + y² = 144 {x² - 4y² = 64
four solutions: (±8√2, ±4)
Find the inverse of the function and determine whether the inverse is a function. f(x) = 19x²
f⁻¹(x) = ±√(x/19), f⁻¹(x) is not a function.
The graph of f(x) = x² is stretched vertically by a factor of 3, translated 4 units to the right, and translated 8 units downward. Determine the equation of the transformed function, g(x).
g(x) = 3(x - 4)² - 8
Identify the conic section produced by the following diagram. (Evernote)
hyperbola
Write the equation in logarithmic form. 6^-4 = 1/1296
log₆ 1/1296 = -4
Which inequality has the solution shown in the graph? (Evernote)
m + 7 ≥ 9
Describe how the graph of the function y = f(x-3) can be obtained from the graph of y = f(x).
move it 3 to the right
Classify the polynomial by degree and number of terms. Describe the shape of its graph. 7x² + 7x + 7
quadratic trinomial; parabola
Determine the end behavior of the graph of the function f(x) = 3x⁴ - 2x³ + 2x + 2
rises to the left; rises to the right
Describe how the graph of the function y = 15(x + 2) + 4 can be obtained from the graph of y = 3(x + 2).
shift up 4 units, and vertically stretch by a factor of 5
Find the exact value of the sine, cosine, and tangent of -210°.
sin = 1/2 cos = -√3/2 tan = -√3/3 Mathway (Trigonometry) Find the Exact Value
Solve the nonlinear system of equations. {x² + y² = 256 {x + y = 16
two solutions: (0, 16) and (16, 0)
Find the variance and the standard deviation for the following data. 7, 20, 7, 15, 21, 4, 22, 17, 13
variance ≈ 39.78; standard deviation ≈ 6.31 https://www.easycalculation.com/statistics/standard-deviation.php Take the Variance (Population Standard deviation) and Population Standard deviation
Which set of characteristics matches the given standard equation and parabola graph? (Evernote) x = 1/24 y²
vertex: (0, 0) directrix: x=-6
Use substitution to determine which of the given linear expressions is a not a factor of 3x⁴ - 16x³ - 15x² + 88x + 60
x + 5 Mathway (Algebra) Factor
Solve the equation for x. Write the exact solution and the approximate solution to the nearest hundredth, when appropriate. In(8x - 6) = 3
x = (3/4) + (e³/8) x ≈ 3.26 Exponential form e³= 8x - 6
solve for x. 3x - 3 = x - 9
x = -3
Write the system of equations as a matrix equation. Then solve the system, if possible, by using a matrix equation. If not possible, classify the system. {x + 2y + z = -17 {3x + 7y + 2z = -56 {x - y + 2z = 0
x = -4, y = -6, z = -1
If x = 78 when y = 130 and x varies directly as y, then find x when y = 190.
x = 114 when y = 190 Formula: y = ky(solving for y) or x = ky(solving for x) Substitute 78 = k(130) Divide by 130 on both sides k = 0.6 x = 0.6(190) x = 114
Write the pair of parametric equations as a single equation in x and y. {x = 8t² -3 [y = 6t - 7
x = 8((y + 7)/6)² - 3
Solve the equation 3^4x = 27^x+3
x = 9
Write an equation that can be used to solve the problem. Then answer the question asked. A group of college students are volunteering for Habitat for Humanity during their spring break. They are putting the finishing touches on a house they built. Working alone, Dale Horton can paint a certain room in 3 hours. Kathy Garcia can paint the same room in 9 hours. How long will it take them working together to paint the room?
x/3 + x/9 = 1; 2.25 hr
Find the standard equation for the ellipse, using the given characteristics. vertices: (0, ±8) foci: (0, ±√55)
x²/9 + y²/64 = 1
Write the product as a polynomial in standard form. (x - 6)(x + 5)(x - 4)
x³ - 5x² - 26x + 120
Gupta threw a baseball off a cliff into an open field 50 feet below. The chart gives the horizontal distance, x (in feet), the baseball traveled from Gupta and the height, y (in feet), of the baseball above the field. (Evernote) distance, x | 8 | 18 | 33 | 43 | height, y | 65 | 76 | 79 | 71 | Choose the quadratic equation that best fits the baseball's trajectory from Gupta to the open field below.
y = -0.0383x² + 2.143x + 50
Write an equation in slope-intercept form of the line that passes through (- 5, 4) and is parallel to the graph of y = -4x + 1.
y = -4x -16
Find a quadratic function that fits the set of data points exactly, in the form y = ax² + bx + c with values of a, b, and c to two decimal places. (1, 4.1), (5, 2.3), (11, 4.9)
y = 0.09x² - 0.98x + 4.99 Mathway: Find the Equation of the Parabola
Write an equation in slope-intercept form for a line that passes through the given pair of points. (4,-7), (-2,-10)
y = 1/2x - 9 Solution Slope Formula: slope(m) = (y2 - y1) / (x2 - x1) Substitute and Solve m = (-10 +7) / (-2 - 4) m = (-3) / (-6) m = 1/2 or 0.5 Slope intercept form: y = mx + b Pick one of the two points (4,-7) or (-2,-10), substitute and solve -7 = 1/2(4) + b -7 = 2 + b -7 - 2 = b -9 = b Final Answer y = 1/2x - 9
Write the pair of parametric equations as a single equation in x and y. {x = 8t - 4 {y = 6t² - 2
y = 6((x+4)/8)² - 2 Solution x = 8t - 4 Add 4 to both sides x + 4 = 8t Divide by 8 on both sides t = (x + 4)/8 Substitute y = 6((x+4)/8)² - 2
A company guarantees customer satisfaction on the purchase of a product, or the company will refund the purchase price of the product. Previous experience has shown that 6% of all purchases are returned. What is the probability that no more than 1 of the next 7 purchases will be returned?
≈ 0.938 ((₇C₁)(0.06)¹(1 - 0.06)⁶) + ((₇C₀)(0.06)⁰(1 - 0.06)⁷) ≈ 0.938
The class average on a math test was 77.5 and the standard deviation was 5.8. Find the z-score for a test score of 83, and the percentage of the class who scored below 83.
≈ 0.95; 82.89% Solution z-score z = (x - x̅) ÷ σ z = (83 - 77.5) ÷ 5.8 = 0.948 → 0.95 (Graphing calculator) 2nd→VARS→DRAW→1:ShadeNorm(-10⁹⁹, 83, 77.5, 5.8)→Enter = 0.83 or 83%
Evaluate the expression to the nearest thousandth. If the expression is undefined, write undefined: e^¼
1.284
A server at a restaurant kept track of the number of requests for the daily special, and the time of day the request was made. The data is displayed below. (Evernote)
Incorrect: It is a linear model with a positive correlation.
Find the mean, the median, and mode of the data set: 14, 6, 21, 16, 16, 7, 20, 19, 7
Mean: 14 Median: 16 Mode: 7, 16
Ms. Kim, an analyst at Multi-Fastener Corp, has developed a model to forecast the company's total production of metal fasteners. Under this model, production, P, is equal to 4500 times the man-hours spent producing machine screws (S) plus 2900 times the man-hours spent producing nuts and bolts (P). Manpower constraints mean that total number of man-hours per year cannot exceed 44,000; logically, the number of man-hours assigned to either task cannot be negative. Express the model in mathematical form.
P = 4500S + 2900B {S + B ≤ 44,000 {S ≥ 0 {B ≥ 0
Find the inverse of the following matrix (if it exists): [2 -13 -6] [0 5 0] [0 -24 0]
The matrix has no inverse.
For the function, use synthetic division and substitution to determine whether the given value is a zero of the function. P(x) = 3x⁴ - 10x³ - 41x² + 68x + 60
Zeroes are 5, 2, -2/3, -3
Solve the quadratic inequality, and graph the solution on a number line. x² - 2x ≥ 24
x ≤ -4 or x ≥ 6
What inequality describes the graph? (Evernote)
x ≤ -5
Find the domain of the radical function. f(x) - √(9-x)
x ≤ 9 Solution f(x) - √(9-x) = 0 Add √(9-x) to both sides f(x) = √(9-x) Mathway: Find the Domain
Solve the radical inequality. √(x + 14) ≤ x - 16
x ≥ 22
Solve the radical inequality. ³√(3x-6) + 6 ≥ 7
x ≥ 7/3
Find the standard equation for the hyperbola with the given characteristics. vertices: (2, 0) and (-2, 0) asymptote: y=1/2x
x²/4 - y² = 1 Put in each answer using Mathway: Find the Vertex Form, till the verices and asymptote matches up
Find the standard equation for the hyperbola with the given characteristics. vertices: (2, 0) and (−2, 0) asymptote: y = 3x
x²/4 - y²/36 = 1
If f(x) = 9 - x² and g(x) = 3 - x, which is the rule of function (f × g)(x)?
x³ - 3x² - 9x + 27
Use elimination to solve the system of equations. {2x - 5y = 1 {5x -2y = -4
(-22/21, -13/21)
Write the system of equations as a matrix equation. Then solve the system, if possible, by using a matrix equation. If not possible, classify the system. {4x + y = -24 {2x + 2y = -18
(-5, -4) You can just solve by using Elimination or Substitution
Use elimination to solve the system of equations. {7x - 3y = -43 {5x + 6y = -47
(-7, -2)
Determine which ordered pair (x, y) is a solution of 3x - y ≤ 15.
(5, 4) Solution Substitute 3(5) - 4 ≤ 15 11 ≤ 15 True
Write an explicit formula for the nth term of the geometric sequence. 7/4, 49/24, 343/144, 2401/864, ...
(Evernote) Solution explicit formula for nth term in a geometric sequence is t(n) = t₁r^(n-1) t₁ = 7/4 and r = 7/6 t(n) = 7/4(7/6)^(n-1)
For the pair of functions, f and g, find (g o f)(x) and (f o g)(x). f(x) = 6 - x, g(x) = x² - 2
(g o f)(x) = x² - 12x + 34 Solution (6-x)² - 2 36 - 6x - 6x + x² - 2 x² - 6x - 6x + 36 - 2 x² - 12x + 34 (f o g)(x) = -x² + 8 Solution -x(x² - 2) + 6 -(x² - 2) + 6 -x² + 2 + 6 -x² + 8
Find the standard equation of a circle with the given radius and center. radius: 7 center: (1, 1)
(x - 1)² + (y - 1)² = 49
Factor the quadratic expression. x² - 11x + 30
(x - 5) (x - 6)
Evaluate the logarithmic expression to the nearest thousandth. log₈ 1/2
-0.333
Find the exact value of sin(- 450° + 300°).
-1/2 Solution Use Sum and Diffrence Identities: sin(A + B) = sinAcosB + cosAsinB sin(450°)cos(300°) + cos(450°)sin(300°) = -1/2 Mathway (Trigonometry) Find the Exact Value
Find all the zeros of the polynomial function. f(x) = x³ + 4x² + 6x + 9
-3, (-1 ±i√11)/2
Evaluate the sum. (Evernote)
-3003
Evaluate the trigonometric expression. Sin⁻¹(-½)
-30° Solution π = 180° -π/6 -180°/6 = 30° Mathway Evaluate (Do not use Find the Exact Value)
A hat contains 22 names, 10 of which are female. If five names are randomly drawn from the hat, what is the probability that at least four female names are drawn?
0.105 ((₁₀C₄ × ₁₂C₁)+(₁₀C₅ × ₁₂C₀))÷(₂₂C₅) ((210 × 12)+(252 × 1))÷(26334) 2772÷26334 = 0.105
Last year, the personal best high jumps of track athletes in a nearby state were normally distributed with a mean of 205 cm and a standard deviation of 13 cm. What is the probability that a randomly selected high jumper has a personal best between 218 and 244 cm? (Evernote)
0.1574 Solution (Graphing calculator) 2nd→VARS→DRAW→1:ShadeNorm(→Enter→75, 90, 85, 7) Between, mean, then standard deviation
A biologist is collecting insects in a field. Beetles represent 67 percent of all the insects that have been collected so far. What is the probability that exactly half of the next 6 insects collected will be beetles?
0.21617 Solution (₆C₃)(0.67)³(1-0.67)³ = 0.21617
The personal savings of the Young Saver Club were normally distributed with a mean of $525 and a standard deviation of $64. What is the probability that a randomly selected saver has an account total between $525 and $589?
0.3413 Solution (Graphing calculator) 2nd→VARS→DRAW→1:ShadeNorm(525, 589, 525, 64)→Enter = 0.3413
A company guarantees customer satisfaction on the purchase of a product, or the company will refund the purchase price of the product. Previous experience has shown that 6% of the purchases are returned. What is the probability that no more than 1 of the next 6 purchases will be returned?
0.954 Solution ((₆C₁)(0.06)¹(1 - 0.06)⁵) + ((₆C₀)(0.06)⁰(1 - 0.06)⁶) = 0.954
Solve the quadratic inequality, and graph the solution on a number line. (Evernote) x² - 4x + 3 < 0
1 < x < 3
In a certain normal distribution of scores, the mean is 30 and the standard deviation is 5.5. Find the z-score corresponding to a score of 37 and find the percentage of the scores that are below 37.
1.27; 89.80% Solution z-score z = (x - x̅) ÷ σ z = (37 - 30) ÷ 5.5 = 1.27 (Graphing calculator) 2nd→VARS→DRAW→1:ShadeNorm(-10⁹⁹, 37, 30, 5.5)→Enter = 0.8984 or 89.80%
Find tan 421°.
1.804 Mathway (Trigonometry) Find the Exact Value
In a bag there are 3 green jelly beans, 2 black jelly beans, and 8 yellow jelly beans. Once a jelly bean is drawn, it is not replaced. Find the probability of randomly drawing a green jelly bean and then a black jelly bean in two consecutive draws.
1/26
Two cards are randomly drawn in succession from a deck of 52 playing cards. Find the probability that the king of spades and any ace are drawn, in that order, without replacement.
1/663 or 0.0015082 Solution 1/52 × 4/51 = 0.0015082
The bowling scores of all of the bowlers in 12 bowling leagues are normally distributed. Their mean score is 201 points, with a standard deviation of 33 points. If one league has 120 bowlers, how many of them are likely to score more than 168 points?
101 Solution (Graphing calculator) 2nd→VARS→DRAW→1:ShadeNorm(168, 10⁹⁹, 201, 33)→Enter = 0.84 .84 or 84% 84% of 120 is 100.8 rounded to the nearest whole number equals 101
Evaluate the expression. 64 × 4² - 2 × 2²
1016
Find the sum of the first 7 terms of the geometric series 8 + 8/3 + 8/9 + 8/27 + ...
11.99 Solution r ≠ 1 r = 8/3 ÷ 8 r = 1/3 & t₁ = 8 S(n) = t₁[(1 - r^n) ÷ (1 - r)] S₇ = 8[(1 - (1/3)⁷) ÷ (1 - 1/3)] S₇ = 11.99
In Sean's school there are 96 families which have 4 children. The circle graph shows the probability of each combination of girls and boys in a family with four children. Use the circle graph to predict the probability that one of these 96 families, chosen at random, will only have children of the same sex. (Evernote)
12.5%
Kozinski played in 16 basketball games this season, Hussein played in 32 games, and Johnson played in 55 games. Kozinski averaged 6 points and 2 rebounds per game, Hussein averaged 8 points and 11 rebounds, and Johnson averaged 18 points and 16 rebounds. Multiply the following matrices to get the total number of points scored and the total number of rebounds made, by all three players combined. [ 6 2 ] [16 32 55] [ 8 11] [18 16]
1342 points, 1264 rebounds
Find the three positive geometric means between 6 and 1875/8.
15, 75/2, 375/4 Solution t₁=6 and t₅=1875/8 t(n) = t₁r^(n-1) t₅ = 6r⁴ 1875/8 = 6r⁴ r = 2.5 t₂ = 6(2.5)¹ 15 t₃ = 6(2.5)² 37.5 or 75/2 t₄ = 6(2.5)³ 9375 or 375/4
Find the sum of the infinite geometric series, if it exists. 3 - 9/5 + 27/25 - 81/125 + 243/625 - ...
15/8 Solution (9/5) ÷ 3 = 0.6 r = 0.6 (common ration) t₁ = 3 S = t₁ ÷ (1 - r) S = 3 ÷ (1 - 0.6) S = 3 ÷ 0.4 S = 7.5 or 15/8
Find the four arithmetic means between -3 and 102.
18, 39, 60, 81 Solution t(n) = t(1) + (n - 1)d 102 = -3 + (6 -1)d 102 = -3 + 5d 105 = 5d d = 21 Substitute t(2) = -3 + (2 - 1)21 -3 + (1)21 -3 + 21 18 t(3) = -3 + (3 - 1)21 -3 + (2)21 -3 + 42 39 And so on
For ∆ ABC, find the measure of ∠A to the nearest degree. (Evernote)
19° Solution Use Law of Cosines for "c" c = √(a² + b² - 2ab cos C) c = 37 a = 12, b = 35, and c = 37 Use Law of Sines sin A/a = sin C/c sin A/12 = sin(90°)/37 sin A = 12sin(90°)/37 sin A ≈ 12/37 Convert from radians to degrees ∠A = 18.58° → 19°
Find the sum of the infinite geometric series, if it exists. (Evernote) ∑3(-½)^(k-1)
2 Solution ∑3(-½)^(k-1) t₁ = 3(-½)⁰ 3 t₂ = 3(-½)¹ -1.5 r = t₂/t₁ r = -1.5 ÷ 3 r = -½ S = t₁/(1 - r) S = 3/(1 - (-½)) S = 3/1.5 S = 2
An airplane, flying at an altitude of 3.21 miles above the ground, has a horizontal speed of 350 miles per hour and is descending at a rate of 14 miles per hour. Use parametric equations to find the airplane's altitude above ground after it has traveled 11 miles, as measured along the ground.
2.77 mi Solution x(t)=350t y(t)=3.21-14t Solve either equation for "t" x=350t t=x/350 Substitute y=3.21-14(x/350) Substitute for "x", x=11 y=3.21-14(11/350) y=2.77
Find the sum of the geometric series 0.2 + 0.02 + 0.002 + . . . given the formula S=a/(1 - r), where "a" is the first term, "r" is the common ratio, and "S" is the sum.
2/9 Solution S = 0.2/(1 - 0.1) S = 0.222... or 2/9
Three cards are drawn in succession and without replacement from a standard deck of 52 cards. How many sets of three cards are possible?
22,100 Solution 52 choose 3 ₅₂C₃ = 22,100
A circular, rotating, serving tray has 5 different desserts placed around its circumference. How many different ways can all of the desserts be arranged on the circular tray?
24 Solution (n-1)! (5-1)! 4! = 4 × 3 × 2 × 1 = 24
How many different arrangements can be made using all of the letters in the word GAME?
24 Solution ₄P₄
Expand the binomial raised to a power. (3a - b)⁵
243a⁵ - 405a⁴b + 270a³b² - 90a²b³ + 15ab⁴ - b⁵ Mathway: Expand Using the Binomial Theorem
A 50-row theater has 30 seats in the front row. The second row has 31 seats. If each row has one more than the row in front of it, how many seats are there in the theater?
2725 Solution t₁ = 30 and d = 1 t(n) = t₁ + (n-1)d t₅₀ = 30 + (49)1 t₅₀ = 79 S = n/2[2t₁ + (n - 1)d] S = 50/2[2(30) + 49] S = 25(60 + 49) S = 25(109) S = 2725
Use either the Law of Sines or the Law of Cosines to solve for "a" to the nearest tenth. (Evernote)
28.8 Solution ∠A + 61° + 76° = 180° ∠A = 43° Use Law of Sines a/sin(43°) = 41/sin(76°) a = 41sin(43°)/sin(76°) a = 28.8
Teesha is in the bowling club. There are 33 students in the club. Five of them will be picked at random to attend an awards banquet. What is the probability that Teesha will not be randomly chosen to attend the banquet?
28/33 Solution Probability of getting chosen is 5/33, so the the probability of not getting chosen is 33 - 5 = 28
The formula for estimating the number, N, of a certain product sold is N = 7400 In(7t + 3), where t is the number of years after the product is introduced. What is the expected number of sales 7 years after the product is introduced? Round to the nearest whole number.
29,239
Find the real zeros of the function. Give approximate values to the nearest hundredth , if necessary. f(x) = x⁴ - x³ - 16x² + 10x + 60
3, -2, ±3.16
Refer to ∆ ABC below to find the indicated value listed. Find the exact value and the value rounded to the nearest ten-thousandth, if necessary. Find tan x (Evernote)
3/4 or 0.75 Solution tan x = opp./adj. 24/32 = 0.75
Find the amplitude, the period, and the frequency of the graph. Then write an equation for the sine function for the graph.
3/4, 180°, 2, y = 3/4 sin 2x Graph all multiple choices and pick the one that fits
Use the Law of Cosines to solve for "A" to the nearest tenth of a degree. (Evernote)
35.3° Law of Cosines cos A = (b² + c² - a²)/(2bc) cos A = (43² + 50² - 29²)/(2 × 43 × 50) cos A = 0.8158 cos⁻¹(0.8158) = 35.3° Just do this ↓ ∠A = cos⁻¹((b² + c² - a²)/(2 × 43 × 50)) ∠A = 35.3°
Abbiville is a small town that has been steadily growing since 1960. Use the table below to create a linear equation that estimates Abbiville's population over time. What will the population be in 2014 if the growth remains constant? (Evernote)
364 people
When Spheres-R-Us ships bags of golf balls, each bag must be within 5 balls of 370. Identify the inequality which results in an acceptable number of golf balls in each bag.
365 ≥ 375
Write the first five terms of the sequence defined by the given recursive or explicit formula. (Evernote)
4, -3, 31, -167, 959
Suppose you mix-up the cards below and choose one without looking. What is the probability of selecting neither "E" nor "G"? H E H E G W I
4/7 Solution Probability of selecting "E" or "G" is 3/7. 7/7 - 3/7 = 4/7
A lunch menu consists of 4 different kinds of sandwiches, 2 different kinds of soup, and 5 different drinks. How many choices are there for ordering a sandwich, a bowl of soup, and a drink?
40 4 × 2 × 5 = 40
Which of the following is an example of the Commutative Property of Addition?
5 + 2 = 2 + 5
Use the quadratic formula to solve the equation. 6x² - 24x - 30 = 0
5, -1
Solve log₃(x+4) - log₃(x-4) = log₃5 for x
6
Simplify the sum, difference, product, or quotient. Assume that the value of any variable is positive ³√(-27x) - 4 ³√(x⁴) + 9 ³√(x) + 9x ³√(x)
6 ³√(x) + 5x ³√(x)