ACT PREP
midpoint formula
(x₁+x₂)/2, (y₁+y₂)/2
Volume of a pyramid
1/3Bh
In the following figure, each circle is tangent to the other two circles, and the two shaded circles are identical to each other. What is the ratio of the shaded region to the non-shaded region? A. 1:1 B. 4:5 C. 5:4 D. 4:π E. π:4
A Let x equal the radius of a shaded circle, and then plug this value into the formula for the area of a circle: A= πr^2 = πx^2 The area of the shaded region is two of these circles, so it's 2πx2. The shaded circles are identical, so the radius of a shaded circle is half that of the large circle. As a result, the radius of the large circle is 2x. Use the formula for the area of the circle: A= πr^2 = π(2x)^2 = 4πx^2 You already know that 2πx2 of this area is shaded, so the other 2πx2 is not. Thus, the shaded and non-shaded regions are the same size. In other words, they're in a 1:1 ratio.
A two-digit number from 10 to 99, inclusive, is chosen at random. What is the probability that this number is divisible by 5? A. 1/5 B. 2/9 C. 19/90 D. 18/91 E. 19/91
A Ninety numbers exist in the range from 10 to 99, and 18 of them are divisible by 5. Place these two numbers into the formula for probability: Probability = target outcomes/total outcomes = 18/90 = 1/5
On an xy-graph, what is the length of a line segment drawn from (-3, 7) to (6, -5)? A. 15 B. 16 C. 17 D. 18 E. 20
A Plug the coordinates for the two points into the distance formula: d = √[( x₂ - x₁)² + (y₂ - y₁)²]
If the first day of the year is a Monday, what is the 260th day? A. Monday B. Tuesday C. Wednesday D. Thursday E. Friday
A Sketch out a little calendar until you see a pattern: Day 1 is Monday, 2 is Tuesday, 3 Wednesday, 4 Thursday, 5 Friday, 6 Saturday, 7 Sunday, 8 Monday, and so on. Notice that Sundays are always multiples of 7. Pick a multiple of 7 close to 260, such as 259. That means Day 259 is a Sunday, so Day 260 is a Monday.
Let f(x) = x^(2)+ 10x+ 2. If g(x) is a transformation that moves f(x) both one unit up and one unit to the right, then g(x) = A. x^(2)+ 8x- 6 B. x^(2)+ 9x+ 3 C. x^(2)+ 10x- 6 D. x^(2)+ 11x+ 3 E. x^(2)+ 12x+ 6
A The function g(x) is the transformation that moves f(x) = x^(2)+ 10x+ 2 one unit up and one unit to the right. To move one unit up, add 1 to the entire function. And to move one unit to the right, substitute x- 1 for xin the function. Thus g(x) = f(x- 1) + 1 Thus, you need to substitute x- 1 for xthroughout the f(x) and to add 1 to f(x): g(x) = (x- 1)^(2)+ 10(x- 1) + 2 + 1 Now simplify: = (x- 1)(x- 1) + 10(x- 1) + 2 + 1 = x^(2)- 2x+ 1 + 10x- 10 + 2 + 1 = x^(2)+ 8x- 6
If p percent of 250 is 75, what is 75% of p? A. 22.5 B. 25 C. 75 D. 225 E. 250
A Translate the statement "p percent of 250 is 75" into an equation and solve for p: p(0.01)(250) = 75 2.5p= 75 p= 30 Thus, 75% of 30 = 22.5.
Which of the following is equal to sin x sec x? A. tan x B. cot x C. cos x tan x D. cos x csc x E. cot x csc x
A sec x = 1/cosx sin x sec x = sinx/cosx sinx/cosx = tanx
On an xy-graph, three corners of a parallelogram are located at (3, 3), (4, - 4), and (-2, -1). Which of the following points could be the remaining corner? A. (8, 0) B. (8, -1) C. (-1, 9) D. (-3, 6) E. (-5, 7)
A good way to begin is to draw a picture showing the three points given, including possible places where a fourth point would form a parallelogram. This figure shows three possible points for the remaining corner of the parallelogram: A, B, and C.To find the exact coordinates of these three additional points, choose any of the given points and count up and over (or down and over) to a second given point. Then, starting from the third given point, count the same number of steps up and over (or down and over) and label the point where you end up. By this method, you find that A = (-3, 6), B= (9, 0), and C= (-1, -8). Only point A is listed as an answer.
Area of a trapezoid
A=1/2h(b1+b2)
Area of a circle
A=πr²
Al bikes a trail to the top of a hill and back down. He bikes up the hill in m minutes, then returns twice as quickly downhill on the same trail. What is the total time, in hours, that Al spends biking up the hill and back down? A. m/60 B. m/40 C. m/30 D. 3m/2 E. 2m
B Al's biking time going up the hill was m minutes, and because he went down the hill twice as fast, his time going down was 1/2m. His total time going up and down the hill was therefore m + 1/2m or 3/2m. If you pick choice (D), be careful-you're not done here! The variable m represents the time in minutes, and the question asks for the time in hours; therefore, you need to divide the total value by 60: (3/2m)/60 = m/40
If (a/c) - (a/b) = (b-c)/a , with a > 0, b> 0, and c> 0, what is the value of a in terms of band c? A. b-c B. √(bc) C.√(b-c) D.√(b-c)/(bc) E. √bc/(b-c)
B Begin by multiplying all three terms by a common denominator of abc to get rid of the fractions: When the denominators are canceled out, the result is the following equation: a2b- a2c= bc(b- c) Factor out a2 on the left side of the equation: a2(b- c) = bc(b- c) Now divide both sides of the equation by (b- c) and cancel: a^2 = bc a = √(bc)
In the following figure, the base of the pyramid has the same area as the base of the cylinder, and the cylinder is twice the height of the pyramid. What is the ratio of the volume of the pyramid to the volume of the cylinder? A. 1 to 3 B. 1 to 6 C. 2 to 3 D. 3 to 2 E. 6 to 1
B Let x equal the area of the base of the pyramid and y equal the height of the pyramid. Plug these values into the formula for the volume of a pyramid: v = 1/3(bh) = 1/3(xy) So x equals the area of the base of the cylinder and 2y equals the height of the cylinder. Now plug these values into the formula for the area of a cylinder, substituting x for πr^2: v = πr^(2)h = x(2y) = 2xy Make a fraction using the area of the pyramid as the numerator and the area of the cylinder as the denominator, and then cancel out xyin the numerator and denominator: (1/3(xy))/(2xy) = (1/3)/2 Multiply the numerator and the denominator by 3 to eliminate the extra fraction: 1/6
If 0° ≤ x° ≤ 180° and 4cos^(2)x = 1, then x= ? A. 0° B. 60° C. 90° D. 150° E. 180°
B Since 4cos^(2)x = 1, cos^(2)x = 0.25 and cosx = 0.5. You want to know x, the degree measure whose cosine is 0.5. A scientific/graphing calculator can help you calculate that: the cos^(-1) key will tell you the degree measure that yields the cosine you give it. cos^(-1)(0.5) = 60°, so choice (B) is correct. If you prefer, you can try each of the answers in your scientific/graphing calculator. When you plug in choice (B), you can find that cos60° = 0.5, so cos^(2)60° = 0.25, and 4cos^(2)60° = 1. Make sure your calculator is in degree mode!
In the following figure, the area of the shaded region is 20% of the area of the whole circle centered at P.The angle shown measures d degrees. What is its measurement in radians? A. 1/5π B. 2/5π C. 4/5π D. 2/15π E. 4/15π
B The area of the shaded region is 20% of the whole circle, so d° is 20% of 360°: d= (0.2)(360) = 72 Use the formula for converting degrees to radians, and plug in 72 for degrees and r for radians: 180/π = degrees/radians 180/π = 72/r Cross-multiply and solve for r: r = 2/5π
In the following figure, if the dimensions of the trapezoid are as shown and the area of the trapezoid is 144, what is the value of x? 6x ⏢ 4x 12x F. 2 G. 3 H. 4 J. 6 K. 8
Begin by plugging the height and the two bases into the formula for the area of a trapezoid: A=1/2h(b1+b2) 144 = 1/2(4x)(6x+12x) 2=x
Ansgar is writing a novel. He writes seven days a week. On each of those days he writes for at least 4 hours but never more than 8 hours. Last week, he wrote for exactly 46 hours. What is the maximum number of days on which he could have written for 8 hours? A. 2 days B. 3 days C. 4 days D. 5 days E. 6 days
C Ansgar writes for at least 4 hours a day, so in 7 days he writes for at least 28 hours (because 4 ×7 = 28). On any day that he wrote for 8 hours, he would have written for an additional 4 hours over the minimum. Thus, the week he wrote 46 hours, he wrote for an extra 18 hours (because 46 - 28 = 18). As a result, he could have written for an additional 4 hours on no more than 4 different days. For example, here's one possible schedule: Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Total 8 8 8 8 6 4 4 = 46
Anderson has a phone plan that charges a monthly rate of $50 for the first 1,000 minutes plus $0.25 for each additional minute. Which of the following functions models Anderson's plan for all m> 1,000, with mas the number of minutes per month and f(m) as the monthly charge? A. f(m) = 0.25m B. f(m) = 0.25m+ 50 C. f(m) = 0.25m- 200 D. f(m) = 0.25m- 950 E. f(m) = 0.25m+ 1,000
C For m > 1,000 — that is, beyond 1,000 minutes of usage — the plan charges a usage fee based on the number of minutes used plus a flat rate of $50. With this information, you can create the following function: f(m) = Usage fee + 50 The usage fee is $0.25 per minute; however, the first 1,000 minutes are already paid for. So the usage fee is 0.25(m- 1,000). Plug this value into the preceding function and simplify: f(m) = 0.25(m- 1,000) + 50 f(m) = 0.25m- 250 + 50 f(m) = 0.25m- 200
In the following figure, line a and line b are parallel and pass through the points shown. What is the equation for line b? Line A: (0, 5), (8,0) Line B: (0,-3) A. y = (5/8)x-3 B. y = -(5/8)x+5 C. y = -(5/8)x-3 D. y = (8/5)x+5 E. y = -(8/5)x-3
C Line a goes "down 5, over 8," so its slope is -5/8. Line b is parallel, so it has the same slope and has a y-intercept of -3. Plug these numbers into the slope-intercept form to get the equation: y = mx+b y = -(5/8)x-3
In the following figure, what is the length of jk? J=(-12,-7) K=(12,3) A. 24 B. 25 C. 26 D. 27 E. 28
C Plug the values (-12, -7) and (12, 3) into the distance formula: d = √[( x₂ - x₁)² + (y₂ - y₁)²] =26
Consider all positive integer values a and b such that the product ab = 8. For how many values does there exist a positive integer c that satisfies both 2^a = c and cb = 256? A. Infinitely many B. 6 C. 4 D. 2 E. 0
C Since you need a and b such that ab = 8, your four possible pairs are a = 1, b = 8; a = 2, b = 4; a = 4, b = 2; a = 8, b = 1. There are four values of c that work in both expressions.
A password for a computer system requires exactly 6 characters. Each character can be either one of the 26 letters from A to Z or one of the ten digits from 0 to 9. The first character must be a letter and the last character must be a digit. How many different possible passwords are there? A. less than 10^7 B. between 10^7 and 10^8 C. between 10^8 and 10^9 D. between 10^9 and 10^10 E. more than 10^10
C The first character must be a letter, so 26 possibilities exist for this character. The last (6th) character must be a digit, so 10 possibilities exist for this one. The remaining four characters can be either a letter or a digit, so 36 possibilities exist for each of these. The following chart organizes this information: 1st 2nd 3rd 4th 5th 6th 26 36 36 36 36 10 Multiply these results: 26 ×36 ×36 ×36 ×36 ×10 = 436,700,160 This result has 9 digits, so it's between 10^8(100,000,000) and 10^9(1,000,000,000).
inside top angle 100 inside x angle __△__ outside y angle In the following figure, what is the value of yin terms of x? A. x+ 80 B. 80 - x C. x+ 100 D. x- 100 E. 100 - x
C The question gives you two of the three interior angles of the triangle: x° and 100°. The remaining interior angle is supplementary with y°, so it's (180 - y)°. Thus, you can make the following equation: x+ 100 + (180 - y) = 180 Solve for yin terms of x: x+ 100 + 180 - y= 180 x+ 100 - y= 0 x+ 100 = y
The ratio of adults to girls to boys on a class field trip was 1:4:5. If the trip included 6 more boys than girls, how many adults were with the group? A. 3 B. 4 C. 6 D. 8 E. 12
C The ratio of girls to boys was 4 to 5, so write the ratio like this: girls/boys = 4/5 If you let g equal the number of girls on the trip, you know that the number of boys was g + 6. Plug these values into the ratio: g/(g+6)=4/5 Cross-multiply and solve for g: 5g = 4(g+ 6) 5g= 4g+ 24 g= 24 So now you know that 24 girls went on the field trip, and you're ready to find the number of adults. The ratio of adults to girls was 1:4. That is, the number of adults was 1/4 the number of girls, so you know that 6 adults attended the field trip.
2√5 ∠n° -> ◢ 2 4 In the following triangle, what is the value of sec n? A. √5 B. 2√5 C. √5/2 D. (2√5)/5
C The secant of an angle equals the hypotenuse over the adjacent angle
If the least common multiple of 9, 10, 12, and v is 540, which of the following could be v? A. 18 B. 24 C. 27 D. 36 E. 45
C To begin, notice that 540 isn't a multiple of 24, so you can rule out Choice (B). Now find the least common multiple (LCM) of 9, 10, and 12. The LCM of 9 and 10 is 90, so the LCM of 9, 10, and 12 must be a multiple of 90. Here are the first six multiples of 90: 90, 180, 270, 360, 450, 540 The number 180 is a multiple of 12 as well, so the LCM of 9, 10, and 12 is 180. However, 180 also is a multiple of 18, 36, and 45. So if any of these numbers were v,the LCM of 9, 10, 12, and vwould be 180. As a result, you can rule out Choices (A), (D), and (E), leaving Choice (C) as your only answer.
If (a/b)-(c/d) = 0 and bc = 7, which of the following statements must be true? A. a and bare directly proportional. B. a and c are inversely proportional. C. a and d are inversely proportional. D. b and c are directly proportional. E. c and d are inversely proportional.
C To begin, simplify the problem to remove the fractions: (a/b)-(c/d) = 0 a/b = c/d ad=bc Now substitute 7 for bc: ad=7 Because ad equals a constant, a and d are inversely proportional.
circumference of a circle
C=2πr
If (a^2 + 2ab + b^2)/(a^2 - b^2) = 2a + 2b, what is the value of a- b? A. 1 B. -1 C. 2 D. 1/2 E. -1/2
D Begin by factoring on both sides of the equation: (a+b)(a+b)/(a+b)(a-b) = 2(a+b) Cancel (a+ b) in the numerator and denominator: (a+b)/(a-b)=2(a+b) Next, cancel (a+ b) on both sides of the equation: 1/(a-b)=2 Multiply both sides by (a- b): 1 = 2(a- b) Now divide both sides by 2: 1/2 = a-b
If a sphere is cut by two different planes, dividing it into sections, how many sections is it possible to end up with? A. 2 only B. 2 or 4 only C. 3 only D. 3 or 4 only E. 2, 3, or 4 only
D Imagine cutting an orange: the first slice (one plane) cuts it into two pieces. If you hold those two pieces together and make another slice (the second plane), you cut both of those pieces, thereby creating 4 sections. Eliminate choices (A) and (C). Now, if you repeat this orange-slicing experiment, but your second slice is parallel to the first slice, it cuts a circular slice off only one piece, thereby creating 3 sections, so eliminate choice (B). The only way to keep the orange in two pieces after the first slice is for the second slice to repeat the first exactly. Since the question said the sphere was to be cut by two different planes, this cannot happen; therefore, it's impossible to get only 2 pieces, so eliminate choice (E).
A 25-foot ladder stands against a vertical wall at an angle of n degrees with the ground. If sin n = 4/5 , how far is the base of the ladder from the wall? A. 12 B. 13 C. 14 D. 15 E. 16
D Note that the question asks you to find the distance from the base of the ladder to the wall, which is the adjacent side of this triangle. Begin by using the sine of n,which is the ratio of the opposite side over the hypotenuse: sin n = O/H = 4/5 O/25=4/5 O=20 Now use the Pythagorean theorem to find the length of the adjacent side: 20^(2) + b^(2) = 25^(2) b=15
A rectangular box has two sides whose lengths are 3 centimeters and 9 centimeters and a volume of 135 cm^3. What is the area of its largest side? A. 27 cm^2 B. 36 cm^2 C. 39 cm^2 D. 45 cm^2 E. 48 cm^2
D The box has dimensions of 3 and 9 and a volume of 135, so plug these values into the formula for the volume of a box: V =lwh 135 = (3)(9)h 5 = h So the remaining dimension of the box is 5. The two longest dimensions are 5 and 9, so the area of the largest side is 5 ×9 = 45.
Sebastian bought a meal at a restaurant and left a 15% tip. With the tip, he paid exactly $35.19. How much did the meal cost without the tip? A. $28.98 B. $29.91 C. $30.15 D. $30.60 E. $30.85
D The tip is a percent increase of 15%, which is 115%. Let x equal the price before the tip. Thus, 115% of this price equals $35.19: 1.15x= 35.19 Divide both sides by 1.15: x = (35.19/1.15)=30.60
Which of the following is the domain of the function: f(X) = (3-x)/√(x^(2) - 9) A. -3 > x> 3 B. -3 ≤x≤3 C. -3 ≤x< 3 D. x< -3 or x< 3 E. x≤-3 or x≥3
D The value inside the radical must be greater than or equal to 0. Thus, if x= 0, the value inside the radical is: x^(2) -9 = 0^(2)- 9 = -9 This value is impossible, so you can rule out Choices (A), (B), and (C). Additionally, the value of the denominator can't equal 0. If x= 3, the value of the denominator is 0. This value is also impossible, so Choice (E) is ruled out as well, leaving Choice (D) as the correct answer.
If 3x+ 5y= 4, which of the following is equivalent to the expression (6x+ 10y)(100x+ 100y)? A. 100x+ 100y B. 200x+ 200y C. 400x+ 400y D. 800x+ 800y E. 1,600x+ 1,600y
D To begin, factor a 2 out of (6x+ 10y): (6x+ 10y)(100x+ 100y) = 2(3x+ 5y)(100x+ 100y) Now substitute 4 for 3x+ 5y and distribute: 2(4)(100x+ 100y) = 8(100x+ 100y) = 800x+ 800y
What is the formula of a line that is perpendicular to (1/3)x +9 and includes the point (3, 4)? A. y = (1/3) + 5 B. y = (-1/3)x +13 C. y= 3x+ 5 D. y= -3x+ 5 E. y= -3x+ 13
E Any line perpendicular to (1/3)x +9 has a slope of -3. So you can rule out Choices (A), (B), and (C). Plug this number into the slope-intercept form, along with the x- and y-coordinates for the point (3, 4): y= mx+ b 4 = -3(3) + b 4 = -9 + b 13 = b Now plug the slope m= -3 and the y-intercept of 13 into the slope-intercept form to get the formula of the line: y= -3x+ 13
Anne and Katherine are both saving money from their summer jobs to buy bicycles. If Anne had $150 less, she would have exactly as much as Katherine. And if Katherine had twice as much, she would have exactly 3 times as much as Anne. How much money have they saved together? A. $300 B. $400 C. $450 D. $625 E. $750
E If Anne had $150 less, Katherine would have three times more than Anne. Make this statement into an equation and simplify: 3(a- 150) = k 3a- 450 = k And if Katherine had twice as much, she would have three times more than Anne: 2k= 3a Substitute 3a - 450 for k into the last equation and solve for a 2(3a- 450) = 3a 6a- 900 = 3a -900 = -3a 300 = a Now substitute 300 for a into the same equation and solve for k: 2k= 3(300) 2k= 900 k= 450 Thus, together Anne and Katherine have 300 + 450 = 750, so the right answer is Choice (E).
In the following figure, ABCD is a square and is a diameter of the circle centered at O.If the area of the square is 100, what is the area of the shaded region? (thing below is a circle that is halfway in the box and halfway outside of the box) ◠ ◛ A. 25π B. 50π C. 100π D. (5π)/2 E. (25π)/2
E The area of the square is 100, so each side of the square is 10 (because 10^2= 100). The diameter of the circle is 10, so its radius is 5. Plug this value into the area formula for a circle: A = π(r^2) A = π(5^2) A = 25π The shaded region is half of this area, so (25π)/2
The figure below is a pentagon (5-sided figure). Suppose a second pentagon were overlaid on this pentagon. At most, the two figures could have how many points of intersection? A. 1 B. 2 C. 5 D. 10 E. Infinitely many
E The key phrase in this question is at most. It's possible that the second pentagon is the same size and is laid directly over the original pentagon. Because all points of one pentagon are the same as the other, select choice (E).
If g(x) is a transformation that moves f(x) three units to the right and then reflects it across the x-axis, then g(x) = A. f(-x) + 3 B. f(-x) - 3 C. f(x) + 3 D. -f(x+ 3) E. -f(x- 3)
E The transformation to move f(x) three units to the right is f(x- 3). Then, to reflect this across the x-axis, change it to -f(x- 3).
If pq= 3, then p^3q^4+ p^4q^5= A. 12q B. 7p+ 9q C. 12p+ 20q D. 96p E. 108q
E To begin, notice that the first term of p^(3)q^(4)+ p^(4)q^(5) contains (pq)^3 multiplied by an extra q,and the second term contains (pq)^4 multiplied by an extra q. As a result, you can factor those values out of each respective term to simplify: p^(3)q^(4)+ p^(4)q^(5)= (pq)^(3)q + (pq)^(4)q Now you can substitute 3 for pq and simplify: = (3)^(3)q + (3)^(4)q = 27q + 81q = 108q
If a+ 2b= 2, what is the value of (a/(b-1))+(a/(b-1))^2+(a/(b-1))^3? F. -6 G. 8 H. 10 J. -12 K. 14
F Begin by finding the value of (a/(b-1)): a+2b=2 a=2-2b a=-2(b-1) Now substitute -2 for (a/(b-1)) to get your answer: (-2)+(-2)^2+(-2)^3 = -6
A circular running track is being built in a fenced-in athletic field 100 feet wide and 150 feet long. If a border of 10 feet is needed between the outside edge of the track and the fence, what is the radius of the largest track that can be built? F. 40 G. 45 H. 65 J. 90 K. 110
F Sketch out the rectangular 100-ft by 150-ft field described in the question. The 10-ft border within the field creates a new, smaller rectangle, 80 feet by 130 feet. The largest circle that can fit in this rectangle has a diameter of 80 feet, and therefore a radius of 40 feet. If you picked choice (G), be careful-you may have forgotten to subtract 10 feet on both sides of the track.
Amber decides to graph her office and the nearest coffee shop in the standard (x,y) plane. If her office is at point (-1,-5) and the coffee shop is at point (3,3), what are the coordinates of the point exactly halfway between those of her office and the shop? (You may assume Amber is able to walk a straight line between them.) F. (1,-1) G. (1, 4) H. (2,-1) J. (2, 4) K. (2, 0)
F The point exactly halfway between is another way to describe the midpoint, so plug the two points into the midpoint formula: (x₁+x₂)/2, (y₁+y₂)/2 = (1,-1).
If logˇ(5)x= 2, what is √x ? F. 5 G. 25 H. 32 J. √5 K. 4√2
F To answer this question, you first have to solve for x. To do so, turn the log into an exponent: logˇ(5)x= 2 means 5^2= x Thus, x= 25. The question asks for , so here's your answer: √x = √25 = 5
What is the x-intercept of a line that passes through the point (3, 4) and has a slope of 2? F. -2 G. -1 H. 0 J. 1 K. 2
First, plot the point (3, 4) on a graph. A slope of 2 means "up 2, over 1," so plot this point on a graph, too. The line passes through the x-axis at 1, making the correct answer Choice (J).
Doug, who runs track for his high school, was challenged to a race by his younger brother, Matt. Matt started running first, and Doug didn't start running until Matt had finished a quarter-mile lap on the school track. Doug passed Matt as they both finished their sixth lap. If both boys ran at a constant speed, with Doug running 2 miles an hour faster than Matt, what was Matt's speed? F. 10.5 miles per hour G. 10 miles per hour H. 9 miles per hour J. 8 miles per hour K. 7.5 miles per hour
G Doug runs 2 miles an hour faster than Matt, so let Matt's speed equal x miles per hour. Then Doug's speed equals x + 2 miles per hour. Each lap is one-quarter of a mile, so Doug runs 1.5 miles in the time it takes Matt to run 1.25 miles. Place this information in a chart: Rate Time Distance Doug x+2 1.5/x+2 1.5 Matt x 1.25/x 1.25 The two boys took the same amount of time from the time Doug started, so make an equation by setting the two times in the chart equal to each other, and then solve for x: (1.5/x+2)=(1.25/x) x = 10
In the following figure, the area of the shaded region is 10% of the area of the circle. If the radius of the circle is 10, what is the arc length from P to Q? (Basically a circle, put 10% is shaded (like a piece of pizza missing)) F. π G. 2π H. 4π J. 5π K. 10π
G Plug the radius of 10 into the formula for the circumference of a circle: C=2πr=2(10)π=20π The area of the shaded region is 10% of the area of the circle, so the arc length from Pto Qis 10% of the circumference of the circle. Therefore, the arc length is 2π.
What is the determinant of the matrix | 3 6 | | -1 2 | F. 0 G. 12 H. |0| J. |6| K. |12|
G The determinant of a matrix is a number, not a matrix, so rule out Choices (H), (J), and (K). To determine which of the remaining answers is correct, use the determinant formula ad- bc: (3 ×2) - (-6 ×1) Simplify: = 6 - (-6) = 6 + 6 = 12
The following figure shows a cylindrical tank whose diameter is 3 times the length of its height. The tank holds approximately 231.5 cubic meters of fluid. Which of the following answer choices most closely approximates the height of the tank? F. 2 meters G. 3 meters H. 4 meters J. 5 meters K. 6 meters
G The height of the tank is hand its diameter is 3h,so its radius is 1.5h.The volume of the tank is approximately 231.5 cubic meters. Use 3.14 as an approximation of πand plug these values into the formula for a cylinder: v = πr^2h 231.5 = (3.14)(2.25h^(2))h simplify and solve for h: 231.5 ≈(3.14)(2.25h2)h 231.5 ≈7.065h3 32.767 ≈h3 3.2 ≈h Thus, the height of the tank is closest to 3 meters.
The following figure shows the graph of an equation y= ax^(2)+ bx+ c. Which of the answer choices CANNOT be true? (Upside U moved to the right) F. a< b G. a> b H. a< c J. b< c K. b> c
G The parabola is concave down, so a is negative. It's shifted to the right, so a and b have different signs; therefore, b is positive. It crosses the y-axis above the origin, so c is positive. Therefore, a can't be greater than b.
If the equation x2+ mx+ n= 0 has two solutions, x = kand x = 2k,what is the value of mnin terms of k? A. 2k^2 B. -2k^2 C. -2k^3 D. 6k^3 E. -6k^3
Given that the equation x^2+ mx+ n= 0 has two solutions, x = k and x = 2k,you can work backward to build the original equation. Here's how: x= kx= 2k x- k= 0 x- 2k= 0 Now take the two equations and combine them: (x- k)(x- 2k) = 0 x^2- 2kx- kx + 2k^2= 0 x^2- 3kx + 2k^2= 0 As you can see, m= -3k and n = 2k^2, so mn= -6k^3.
If a square has an area of 64 square units, what is the area of the largest circle that can be inscribed inside the square? F. 4π G. 8π H. 16π J. 64 K. 64π
H Draw yourself a figure to see the relationship between the two shapes. Since the formula for area of a square is A = s^2 (where s is the side length of the square), you can find that a square with area 64 has side length 8, which would also be the diameter of the circle inscribed in this square, meaning the circle's radius would be 4. The formula for the area of a circle is A = πr^2, so A = π(4)^2 = 16π.
If a cube has a volume of kcm^3 and a surface area of 10k cm^2, what is its height in centimeters? F. 1/2 G. 3/4 H. 3/5 J. 4/5 K. 3/10
H Plug k into the formula for the volume of a cube and 10k into the formula for the volume of the surface area of a cube: k = s^3 10k = 6s^2 10s^3 = 6s^2 10s = 6 s = 3/5
Beth got a job painting dorm rooms at her college. At top speed, she could paint 5 identical rooms during one 6-hour shift. How long did she take to paint each room? F. 50 minutes G. 1 hour and 10 minutes H. 1 hour and 12 minutes J. 1 hour and 15 minutes K. 1 hour and 20 minutes
H Six hours equals 360 minutes, so Beth paints 5 rooms in 360 minutes. She paints 1 room in 360 ÷ 5 = 72 minutes, which equals 1 hour and 12 minutes.
The following figure shows a regular octagon inscribed in a circle. The arc length from A to Bis 6π. What is the area of the shaded region of the circle? F. 8π G. 16π H. 24π J. 36π K. 64π
H The arc length from A to B is 6π. The angle from A to B is 3/8 of the circle's total of 360°, which equals 135°. Plug these values into the formula for arc length: arc length = degrees(πr/180) 6π = 135(πr/180) Solve for the radius r: r = 8 Now plug in 8 for r in the formula for the area of a circle: A = π(8)^2 A = 64π The area of the shaded region is 3/8 of this value: 24π
When she chooses a password, Eloise always uses exactly ten different characters: five letters (A, B, C, D, and E) and five numbers (2, 3, 4, 5, and 6). Additionally, she always ensures that no pair of letters is consecutive and that no pair of numbers is consecutive. How many different passwords conform to these rules? F. fewer than 1,000 G. between 1,000 and 10,000 H. between 10,000 and 100,000 J. between 100,000 and 1,000,000 K. more than 1,000,000
H The first character of the password can be any letter or number, so Eloise has ten options. Her second choice must be from the set (letter or number) not yet used, so she has five options. Choice 3 is from the same set as Choice 1, and she has four options left. Choice 4 is the second item from the same set as Choice 2, so she has four options. Choices 5 and 6 are from different sets, each with three options; Choices 7 and 8 are from different sets, each with two options; Choices 9 and 10 are from different sets, each with only one option remaining. You can see this information in the following chart: 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 10 5 4 4 3 3 2 2 1 1 To find the total number of possible passwords, multiply these numbers together: 10 ×5 ×4 ×4 ×3 ×3 ×2 ×2 ×1 ×1 = 28,800
The equation y= ax^b+ c produces the following (x, y) coordinate pairs: (0, 2), (1, 7), and (2, 42).What is the value of abc? F. 10 G. 20 H. 30 J. 40 K. 60
H To begin, plug x= 0 and y= 2 into the equation y= ax^b+ c. Note that the first term drops out: 2 = a(0)b+ c 2 = c Now you can substitute 2 for c in the original equation, giving you y= ax^b+ 2. Next, plug in x= 1 and y= 7. Notice that everything drops out except for the coefficient of the first term: 7 = a(1)^b+ 2 5 = a You can now substitute 5 for a in the equation, giving you y= 5x^b+ 2. Now plug in x= 2 and y= 42: 42 = 5(2)^b+ 2 40 = 5(2)^b 8 = 2^b 2^3= 2^b 3 = b Finally, you can see that abc = (5)(3)(2) = 30.
If x^2- x- 2 > 0, which of the following is the solution set for x? F. x> -1 G. x> 2 H. -1 < x < 2 J. x< -1 or x> 2 K. No solutions are possible.
J Begin by treating the inequality x2- x- 2 > 0 as if it were an equation. Factor to find the zeros: x2- x- 2 = 0 (x+ 1)(x- 2) = 0 x= -1 and x= 2 The graph of this function is a parabola that crosses the x-axis at -1 and 2. This graph is concave up because the coefficient of the x^2 term (that is, a) is positive. The parabola dips below the x-axis when -1 < x < 2, so these values do not satisfy the equation. Therefore, the correct answer is Choice (J). (If you doubt this answer, note that when x= 0, x^2- x- 2 = -2, which is less than 0. So 0 is not in the solution set.)
Pippin the guinea pig is running on her wheel when, due to a manufacturing error, the wheel breaks free of its axis. Pippin remains in her wheel, running in a straight line until the wheel has rotated exactly 15 times. If the diameter of the wheel is 10 inches, how many inches has the wheel rolled? F. 75 G. 150 H. 75π J. 150π K. 1,500π
J Since the wheel's diameter is 10, you can find the circumference (C = πd) of Pippin's wheel: C = πd = 10π. This means Pippin's wheel travels 10π inches in one rotation. Since her wheel rotated 15 times, multiply 10π × 15 = 150π.
A sphere is inscribed in a cube with a diagonal of 3√3 ft. In feet, what is the diameter of the sphere? F. 3√2 G.2 H. 2√2 J. 3 K. 3√3
J The formula for the length of the diagonal of a rectangular prism is a^2 + b^2 + c^2 = d^2 where a, b, and c represent the edges of the rectangular prism and d represents its diagonal. In the case of a cube, a = b = c, so the equation can be rewritten as follows: 3a^2 = d^2. In this problem, d = , so 3a^2 = (). Therefore, 3a^2 = (9)(3) and a = 3. Therefore, the length of the edge of this cube is 3. Look at the figure-you can see from the figure that the length of the diameter of the sphere is equivalent t.
If a number sequence begins 1, 3, 4, 6, 7, 9, 10, 12 . . ., which of the following numbers does NOT appear in the sequence? F. 34 G. 43 H. 57 J. 65 K. 72
J The sequence includes all multiples of 3, including 57 and 72, so Choices (H) and (K) are ruled out. The sequence also includes every number that's 1 added to a multiple of 3, which includes 34 (33 + 1) and 43 (42 + 1), so Choices (F) and (G) are ruled out. By the process of elimination, Choice (J) is the correct answer.
Which of the following functions has a range of f(x) ≥4? F. f(x) = |x+ 4| G. f(x) = |x- 4| H. f(x) = |x+ 4| - 4 J. f(x) = |x- 4| + 4 K. f(x) = |x- 4| - 4
J The value of a linear function inside absolute value bars can never be less than 0. Thus, a function with a range of f(x) ≥4 takes this minimum value and adds 4 to it. Therefore, f(x) = |x- 4| + 4 can never be less than 4, so the correct answer is Choice (J).
For all real values of a and b, the equation |a - b| = 5 can be interpreted as "the positive difference of a and b is 5." What is the positive difference between the 2 solutions for a? F. b G. b + 5 H. 2b J. √(b^(2)-25) K. 10
K "Positive difference" means that when you subtract a - b you can get +5 or -5. (That's also what the absolute value indicates.) Solve for a in both cases: a - b = 5, so a = b + 5; and a - b = -5, so a = b - 5. Now subtract these two values: (b + 5) - (b - 5) = b + 5 - b + 5 = 10, choice (K). Alternatively, you could substitute a number for b: let's say b = 2, so |a - 2| = 5. Then solve: a - 2 = 5, so a = 7; and a - 2 = -5, so a = -3. The positive difference is 7 -(-3) = 7 + 3 = 10.
On his first day working out, Anthony did 30 push-ups. On each successive day, he did exactly 3 more push-ups than on the previous day. After completing his push-ups on the 30th day, how many push-ups had he completed on all 30 days? F. fewer than 500 G. between 500 and 1,000 H. between 1,000 and 1,500 J. between 1,500 and 2,000 K. more than 2,000
K Anthony completed 30 push-ups the first day, 33 the second day, 36 the third day, and so on. Make a chart as follows: 1 2 3 4 5 . . . 10 . . . 15 . . . 20 . . . 25 . . . 30 30 33 36 39 42 57 72 87 102 117 To save time adding all these numbers, notice that the total of the first and 30th numbers is 30 + 117 = 147. This total is the same for the 2nd and 29th, the 3rd and 28th, and so on all the way to the 15th and 16th. Therefore, you have 15 pairings of days on which Anthony completed 147 pushups. You can simply multiply to find the total: 147 ×15 = 2,205.
If 49^(3y) = √(7^(y+1)), then y = F. 1/2 G. 1/3 H. 1/5 J. 1/7 K. 1/11
K Begin by squaring both sides of the equation to eliminate the radical. 49^(6y) = 7^(y+1) Now substitute 7^2 for 49 on the left side. 7^(12y) = 7^(y+1) Because the bases are both 7, the exponents are equal. So now you can drop the bases: 12y= y+ 1 Solve for y: y = 1/11
At 10:00, Angela starts from her home and runs at a constant pace to Kathleen's house, which is exactly 2 miles away. Immediately, she and Kathleen turn around and walk back to Angela's house exactly 4 miles an hour slower than Angela ran. When they arrive at Angela's house, the time is 10:45. At what speed did Angela run? F. 6 miles per hour G. 6.5 miles per hour H. 7 miles per hour J. 7.5 miles per hour K. 8 miles per hour
K If you let x be the speed at which Angela ran, you can let x- 4 be the speed at which Angela and Kathleen walked. The distance in each direction was 2 miles, and the total time was 45 minutes, which is of an hour. Place all of this information into a Rate-Time-Distance chart: rate time distance running x 2/x 2 walking x-4 2/(x-4) 2 total 3/4 Adding the Time column, set up the following equation: (2/x)+(2/(x+4)) = 3/4 Use x(x- 4) as a common denominator on the left side of the equation to add the two fractions and simplify: (2(x-4) + 2x)/x(x-4) = 3/4 Now cross-multiply and simplify again: 4(4x - 8) = 3(x2- 4x) 16x- 32 = 3x2- 12x 0 = 3x2- 28x+ 32 Solve the resulting quadratic equation for xusing either factoring or the quadratic formula (I use factoring): (3x- 4)(x- 8) = 0 3x- 4 = 0 x- 8 = 0 The first equation solves for x as a number that's less than 4 (x=4/3), which isn't correct in the context of the question because their speed on the way back would be negative. The second equation solves as x= 8, so the correct answer is Choice (K).
Which of the following is a possible solution for x in terms of k for the equation x = 2k/x+2 F. √(2k) G. √(-2k) H. 1-√(1+2k) J. √(1+2k) + 1 k. √(1+2k) - 1
K Multiply both sides of the equation by x+ 2 to remove the fraction, and then place all terms on one side of the equation: x^2 + 2x = 2k The result is a standard quadratic equation, where a= 1, b= 2, and c= -2k.Use the quadratic formula to solve and simplify.
A function is defined for x and y such that f(x, y) = -2xy + y + x - 4. So, for x = 2 and y = 3, f(2, 3) = -2 × 2 × 3 + 3 + 2 - 4 = -12 + 1 = -11. If x and y are to be chosen such that f(x,y) = f(y,x), then which of the following restrictions must be placed on x and y? F. x > 0 and y > 0 G. x < 0 and y < 0 H. x = y J. xy < 0 K. No restrictions are needed.
K No restrictions are needed because f(x,y) = f(y,x) in all cases. Follow the same rules as the original function, just switch x and y. Because -2yx = -2xy, this is the same result that f(x,y) produced in the question and f(y,x) = -2yx + x + y - 4. Therefore, all values of x and y will result in f(x,y) = f(y,x), so choice (K) is correct.
What is the solution set for x for the inequality | 2x+ 7 | > 11? F. 2 < x< 9 G. -9 < x< 2 H. x< 2 or x> 9 J. x< -2 or x> 9 K. x< -9 or x> 2
K Split the inequality |2x+ 7| > 11 into two inequalities: 2x+ 7 > 11 and 2x+ 7 < -11. Solve both for x: 2x+ 7 > 11 2x+ 7 < -11 2x> 4 2x< -18 x> 2 x< -9 Therefore, x< -9 or x> 2.
Two variables, v and w,are inversely proportional such that when v= 7, w= 14. What is the value of w when v= 2? F. 1 G. 4 H. 14 J. 28 K. 49
K The variables v and w are inversely proportional, so for some constant k,the equation vw= k is always true. Thus, when v= 7 and w= 14: vw= (7)(14) = 98 So k= 98. When v= 2, you can find w like this: vw= 98 2w= 98 w= 49
mean, median, mode, range
Mean:the arithmetic average of a distribution, obtained by adding the scores and then dividing by the number of scores (average) Median: the middle score in a distribution; half the scores are above it and half are below it Mode: the number that occurs most often in a set of data Range: the difference between the highest and lowest scores in a distribution
A square field has an area of 22,500 square feet. To the nearest foot, what is the diagonal distance across the field? F. 150 feet G. 178 feet H. 191 feet J. 212 feet K. 260 feet
The area of the field is 22,500 square feet, so plug this value into the formula for the area of a square to find the length of the side: A = s^2 s^2 = 22,500 150 = s So the side of the field measures 150 feet. The diagonal of a square is the hypotenuse of a 45-45-90 triangle, which has three sides in a ratio of x:x:x√2 . Thus, the length of this diagonal is . (You can also use the Pythagorean theorem to find this result.)
If you plot the equation x^2 + (y- 2)^2= 4 as a circle on a standard xy-graph, what is the area of the circle's region that will lie in Quadrant 1, as shown in the following figure? A. 0 B. π C. 2π D. 4π E. 16π
The formula for a circle with a radius r centered at (a,b) is (x- a)2+ (y- b)2= r2. Thus, x2+ (y- 2)2= 4 has a radius of 2 and is centered at (0, 2), as shown here: A= πr^2=π(2)^2 This circle has a radius of 2, so calculate its total area as follows: Because half the circle is in Quadrant 1, the area of this region is 2π.
Volume of a cube
V=s^3
Which of the following is equivalent to (Y^3)^8? F. y^11 G. y^24 H. 8y^3 J. 8y^11 K. 24y
When raising a number with an exponent to another power, you multiply the exponents; therefore, (y^3)^8 = y^(3×8) = y^24.
Distance Formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
logarithm equation
logˇ(b)X = Y -> X=b^Y
Volume of a box
lwh
Opposite of sin, cos, and tan
sin: cosecant cos: secant tan: cotangent
volume of a cylinder
v = πr^(2)h
y x ◣ z
x: opposite y: hypotenuse z: adjecent