Quantitative Reasoning

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you have a 30% off coupon and after 10% tax, the bill comes to $36. How much was the original bill?

$36 = x(1.10)(0.70) x = $46.75

item was purchased for $378 including 8% tax. what was original cost?

$378/1.08 = $350

insurance bill was $900 which included $500 deductible plus 20% of remaining amount. What was the total cost?

$500 + ($900-$500/0.2) = $2500

evaluate (3,2,-4) * (2,-1,0)

(3*2) + (2*-1) + (-4*0) = 4

average of 72% on 3 tests. What will they need on the 4th test to average 75%?

(72+72+72+x)/4 = 75; x = 84

30-60-90 triangle

- across from 30: x - across from 60: x√3 - across from 90: 2x

45-45-90 triangle

- across from 45: x - across from 90: x√2

domain and range of f(x) = sec(x)

- domain: all real numbers except pi/2 + kpi - range: (-∞, -1] U [1, +∞)

derivative of arccos(x)

- dx/√(1-x²)

what mean and standard deviation would disprove a claim?

- smallest standard deviation (s) - mean (x) that is farthest from the actual mean

cos(π)

-1

i²

-1

i³

-i

sin(0)

0

sin(π)

0

How many real zeros does the derivative of a third degree function have?

0, 1, or 2

cos(0)

1

circle has radius of 4.4 ft. in 5 revolutions, how far does it travel?

1 revolution = 2πr; 5 revolutions = 10πr plug 4.4 into r = 44π

Find standard deviation of 1, 2, 5, 8, 9

1. Find mean of set = 5 2. subtract each number from the mean then square it (16, 9, 0, 9, 16) 3. Find mean of the new set = 10

if f'(x) = 3x² + 2x and (1,3) lies on curve f(x), what is f(2)

1. anti-differentiate to find f(x) ; x³ + x² + c 2. we know that f(1) = 3 so; 3 = (1)³ + (1)² + c; c = 1 3. find f(2); 2³ + 2² + 1 = 13

log(base2)2x + log(base2)x+3 = 3

1. both are log(base2) so you can multiply; log(base2)(2x)(x+3) = 3 2. using log rules --> (2x)(x+3) = 2³ 3. factor and get x = -4, 1 *Logs can't be negative so only answer is 1*

2A = 3B & C = 2B; solve for A in terms of C

1. multiply the first equation by 2 and the second equation by 3 to make the B's equal eachother; 4A=6B & 3C=6B 2. set 6B's equal to eachother; 4A = 3C; A = 3/4C

what is the area bounded by y=2x, the x-axis, and from lines x=1 & x=4?

1. take antiderivative 2. find definite integral from 4 to 1 X² |(4-1) (4)² - (1)² = 15

what is the value of constant "c" so that x-y=c is the equation tangent to f(x) = x² + 3x +4

1. take derivative and set equal to 1 b/c the slope of the line (x-y=c or y=x-c) is 1; 2x+3 = 1; x = -1 2. plug x into f(x) to find y; (-1)² + 3(-1) +4; y = 2 3. plug x and y into tangent equation to find c; -1-2 = -3

ds/dt = 32t - 6 what is the constant in the position function

1. take derivative to find position function; s = 16t² - 6t + c 2. -6t is the initial velocity, c is the initial position

what % of 75 is 1.5?

1.5/75 x 100 = 2%

3^(2x-5) = 1/27

1/27 = 3^-3 3's cancel out 2x-5 = -3; x=1

1/2log25 + log 3/5 + log4

1/2log25 = log25^1/2 = log5 log5 + log3/5 + log 4 = log (5*(3/5)*4) = log12

derivative of ln(x)

1/x

derivative of 1/√x

1/√x = x^(-1/2) derivative = -1/2x^(-3/2) = - 1/ 2x√x

if you have 10 players that can play the 3 outfield positions, how many combinations can you make if you choose 3 of the 10?

10!/7!3!

convert L to cubic centimeters

1000 cc in 1 liter x: 0.2 L x 1000 cc/1 Lite = 20 cc

how many nanograms in a kilogram?

10^9 nanograms in a gram; 10^3 grams in a kilogram 10^9 + 10^3 = 10^12

convert 12 gallons per min to pints per sec

12 gal/1 min x 8 pin/1 gal x 1 min/60 sec = 1.6

a coin is flipped 10 times, what is the probability of exactly 2 heads?

2C10p²q^8 10!/2!x8! = 45 p = q = 1/2 so; (1/2)² x (1/2)^8 = (1/2)^10 = 1024 ans = 45/1024

period of y = a sin bx

2pi/b

circumference of a circle

2πr

f(x) = 2x² + 3 g(x) = |x-5| find (3f - 2g)(2)

3(2*2² + 3) - 2|2-5| (absolute value changes 2-5=-3 to +3 so; 3(11) - 2(3) = 27

volume of a sphere

4/3πr³

15% of what number is 45?

45/0.15 = 300

50 mph is approximately 80 kph. if distance is 30 miles, how many kilometers?

50/80 = 30/x; x = 48

|5x+8| = |x-4|

5x+8=x-4 & 5x+8 = -(x+4) x = -3, -2/3

simplify (6.5x10³/2.5x10^7)

6.5/2.5 - 2.6 10³/10^7 = 10^(3-7) = 10^-4 ans = 2.6x10^-4

what percent falls within +-1 standard deviation

68%

what % falls within +- 2 standard deviations?

95%

equations for degrees F and C

C = 5/9(F-32) F = 9/5C+32

a set of overlapping shingles cover 0.2m² per shingle. How many are needed for a roof that is 5m by 13m?

Find area then divide by 0.2 5*13 = 65/0.2 = 325

John gets 8 miles per gallon, Mary gets 34 miles per gallon. Both use whole # of gallons, what is the shortest mileage?

Find least common multiple of both numbers 8 = 2^3 34 = 2(17) 2^3(17) = 136

given events A and B are independent, find P(A or B)

P(A or B) = P(A) + P(B) - P(A*B)

0.6 probability of losing, 0.25 probability of a tie. what is P of winning or tie?

P(W) = 1 - P(L) - P(T) = 0.15 P( W or T) = P(W) + P(T) = 0.4

If h(t) = 30(sin pi(t)/6) at what rate is h changing when t=2?

Take derivative using chain rule then plug in t=2. Answer= 5pi/2

The mean of a population is 1200, 80% of the data falls within 1.5 standard deviations of mean and lies between 900 and 1500, what is the standard deviation?

Use formula for z-scores. 80% is unnecessary info. Z = x - u / õ 1.5 = 1500 - 1200 / õ õ = 300/1.5 = 200

Given y=x^2-kx+1, where k>_ 0, as k increases, what happens to x and y?

X increases, y decreases

finding domain of function g(x) = √(4-x^2)

acceptable values of x (set equal to zero) 4-x^2 = 0, x= 2, -2 - plug in a value between the two numbers to make sure (ex: 0, (4-0^2) > 0 = true) answer = [-2,2]

probability of guessing correctly is 1/4. What is the probability of guessing on 3 questions and getting at least 1 correct?

at least 1 correct = 1 - probability of guessing wrong 1 - (3/4)³ = 37/64

find the vector perpendicular to (2,-1,3)

choose the answer the gives us 0 when multiplying out ex: (0,3,1) (2, -1, 3) x (0, 3, 1) = (2x0) + (-1x3) + (3x1) = 0

what is the area bounded by the lines y = 2x, y = x, x = 1, and x = 4

definite integral from 4 to 1 of (2x-x)dx; x²/2 (4)²/2 - (1)²/2 = 15/2

derivative of arctan(x)

dx/(1+x²)

derivative of arcsin(x)

dx/√(1-x²)

Which function or x will have a value that is 3 times that if it's derivative regardless of the value of x?

e^(x/3)

factor 15x² + 8x + 1 = 0

factors of 15 that add to get 8 = 3 and 5; use those with the x and use the +1 in the factoring (3x+1)=0 and (5x+1)=0; x = -1/5, -1/3

drove 45 mph to work and 30 mph home. What was the average speed?

find least common denominator of the two numbers = 90 rate = distance / time - To work: 90 miles / 30 mph = 3 hours - Home: 90 miles / 45 = 2 hours (90 + 90) / (2+3) = 36 mph

Find ∑10 i=1 i²-i

for i = n(n+1)/2 for i² = n(n+1)(2n+1)/6 10(11)/2 - 10(11)(21)/6 = 330

if h'(x) = x³ and h(2) = 2h(1) then what is h(x)

h(x) = x^4/4 + C set h(2) = 2h(1) to find C (2)^4/4 + C = 2(1^4/4 + C) C = 7/2 h(x) = x^4/4 + 7/2

equation of plane passing thru (-1,2,3) perpendicular to vector i+j-2k

i(x+1) + j(y-2) + k(z-3) = 0 1(x+1) + 1(y-2) -2(z-3) = 0 x+y-2z = -5

where is f(x) = x/(x²+1) increasing?

increasing when the derivative = 0 [-1, 1]

If the interval [0,2] is split into 4 intervals, with a curve of f(x) = x³+1, what is the total area of the 4 rectangles?

left endpoints are 0, 1/2, 1, 3/2 area = 1/2[f(0) + f(1/2) + f(1) + f(3/2)] = 17/4

ID is 3 letters and 4 numbers. Letters: A, B, C, D can repeat. Numbers: 0, 1, 2, 3, 4, 5 cannot repeat. How many different combinations?

letters: 4³ numbers: (6*5*4*3) multiply together = 4³*(6!/2!)

log 4 = .602 log 6 = .778 find log 40 + log 36

log 40 = log 10 + log 4 = 1 + .602 log 36 = log 6² = 2log6 = 2(.778) 1 + .602 + 2(.778) = 3.158

Log(base 2)√32 + log(base 2)8

log(base 2)√32 = log(base2)32^1/2 = 1/2log(base2)32 = 1/2log(base2)2^5 log(base2)8 = log(base2)2^5 log(base2)2 cancels out on both sides 1/2(5) + 3 = 11/2

log(base5)(x+10) = -1 + log(base5) 125

log(base5) 125 = log(base5)5³ = 5's cancel = 3 log(base5)(x+10) = -1 + 3 = 2 log(base5)(x+10)=2; 5² = x+ 10; x = 15

logb2 = .3 and logb3 = .5, find logb12

logb12 = logb4 + logb3 = logb2*2 + logb3 = 2(0.3)+0.5 = 1.1

given 15 consecutive integers, the relationship between the mean and median

mean = median

applied to 5 schools. admission is independent. P(accepted) = 0.2. What is the probability that she will get accepted into 3 of the 5 schools

nCr*p^r*q^(n-r) nCr = n!/r!(n-r)! n = # of events = 5 r = # of successes = 3 p = prob of success = 0.2 q = prob of losing = 0.8 10(0.2³)(0.8²) = ans

bc√(n^bd)

n^(bd/bc) = n^(d/c)

the mean of 10 numbers is 50 and 50 is one of the numbers. what would happen to the range/standard deviation if 50 is removed?

range would not change, standard deviation would increase

vertical asymptote

set denominator equal to zero

f(x) = ax+4 g(x) = 8x+2 which values of "a" does f(g(x)) = g(f(x)) for all real values of x?

set f(g(x)) = g(f(x)) to find "a" a(8x+2) + 4 = 8(ax+4) + 2 a = 15

If A = 2/3B and C = 3/4B, then what is C in terms of A?

set first equation equal to B; B = 3/2A plug B into equation with C C = 3/4(3/2A) C = 9/8A

When rolling 2 dice, what is the probability the sum is 8?

sum 2 = 1/36 sum 3 = 2/36 sum 4 = 3/36 sum 5 = 4/36 sum 6 = 5/36 sum 7 = 6/36 sum 8 = 5/36 sum 9 = 4/36 sum 10 = 3/36 sum 11 = 2/36 sum 12 = 1/36

Find slope of curve y=2x^3 - 3x^2 + 12x @ x=-1

take derivative, plug in -1 = 24

Frank's hybrid car gets 40 mpg, if he travels 60 mph how far can he go on a tank of 20 gallons?

the 60 mph is excess info that is not needed. 40mph x 20 gallons = 800 miles

what is the slope of: {x,y} = {3,4} + t{1,1}

the pair with the "t" is the slope; 1/1 = 1

how many zeros will 50! end in

to end in a zero we need multiples of 10 = 5x2 only count amount of 5's; 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 = 6 5's multiplied by 2 = 12 zeros

recruiter signed 55 of 66 students. what is the minimum number of recruits they need to sign of the next 24 students to average a 75%?

total students = 66 + 24 = 90 75% of 90 = 90*.75 = 67.5 = 68 students already signed 55, so 68-55 = 13 students

where is f(x) = x³ +3x concave down?

where the second derivative < 0 f'(x) = 3x² + 3 f''(x) = 6x 6x < 0; x < 0 (-∞, 0)

evaluate the indefinite integral dx

x + c (antiderivative of x)

a student took 34 courses; 3 or 4 credits each for a total of 120 credits. How many 3-credit courses did they take?

x = 3 credit y = 4 credit x + y = 34 3x + 4y = 120 multiple first equation by 4 and subtract; x=16, y=18

Average age for a class of 10 students is 29. When 2 students leave, the average is 30. What is the average of the 2 students?

x/10 = 29; x = 290 x/8 = 30; x = 240 subtract; x = 50/2 = 25

f(x) = ³√(4x+3) find f^-1 (x)

y = ³√(4x+3) y³ = 4x+3 x = (y³ - 3)/4 - switch x and y y = (x³ - 3)/4

Given y=3e^2x, the slope of the line tangent is

y'= 6e^2x which is 2y, So the slope is directly proportional to y

Find K if slope thru points (3,k), (k,1) = 2

y1-y2=m(x1-x2) k-1 = 2(3-k) k = 7/3

given the complex number z = 4 + 3i find |z|

| a+-bi | = √(a²+b²) √(16+9) = 5

area of a circle

πr²

volume of a cylinder

πr²h

√((x-2)²) >= 1

√(a²) = |a| |x-2| >= 1 --> x-2>= 1 or -(x-2) >= 1 x >= 3 or x <= 1 ans = (-∞, 1)U(3,+∞)

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