Q: A bakery has 250 lbs of flour. If they use 12% of the available flour each day, how much flour remains after 7 days? (Round to the nearest pound.)

$250 \times (0.88)^7 \approx 110$ lbs

Q: The intensity $I$ of illumination from a light source is inversely proportional to the square of the distance $d$ from the source. If $I = 100$ lumens at $d = 20$ feet, what is the intensity at $d = 30$ feet?

$I = 100 \times \left(\frac{20}{30}\right)^2 = 44.4$ lumens

Question 1

Suppose the weight $w$ (in pounds) of American men is a linear function of their height $h$ (in inches), and the data points $(60, 106)$ and $(72, 180)$ are given. What is the formula for $w$ in terms of $h$?

Accepted Answer

$w = 6.17h - 264.2$

Question 2

A population is growing according to the function $P = 250(1.065)^t$, where $P$ is the population at year $t$. What is the annual growth rate?

Accepted Answer

6.5%

Question 3

A bakery has 250 lbs of flour. If they use 12% of the available flour each day, how much flour remains after 7 days? (Round to the nearest pound.)

Accepted Answer

$250 	imes (0.88)^7 \approx 110$ lbs

Question 4

The total cost $C$ of producing $x$ sneakers is given by $C(x) = 0.01x^2 - 0.2x + 2000$. What is the average rate of change in cost if production increases from $x = 100$ to $x = 150$?

Question 5

A business finds it can sell 12,000 shirts at $\$19$ each, and 3,000 shirts at $\$35$ each. Assuming a linear relationship, what is the slope of the function relating number of shirts $S$ to price $p$?

Accepted Answer

$-500$

Question 6

A ball is thrown from a rooftop and its height is given by $s(t) = -16t^2 + 80t + 96$. When does the ball strike the ground?

Accepted Answer

$t = 6.5$ seconds

Question 7

Money deposited in a bank account doubles every 13 years with continuous compounding. What is the annual interest rate (to the nearest tenth of a percent)?

Accepted Answer

5.3%

Question 8

If $f(x) = 2x^2 - 2x$ and $g(x) = 3x - 1$, what is $f(g(2))$?

Accepted Answer

$f(5) = 2(5)^2 - 2(5) = 50 - 10 = 40$

Question 9

The intensity $I$ of illumination from a light source is inversely proportional to the square of the distance $d$ from the source. If $I = 100$ lumens at $d = 20$ feet, what is the intensity at $d = 30$ feet?

Accepted Answer

$I = 100 	imes \left(\frac{20}{30}ight)^2 = 44.4$ lumens

Question 10

If $250$ mg of a radioactive element decays to $200$ mg in $48$ hours, what is the half-life of the element (to the nearest hour)?

Accepted Answer

66 hours

Question 11

Suppose you invest $\$1000$ at 12% interest compounded annually for 10 years, or at 11.5% compounded continuously for 10 years. Which yields more money?

Accepted Answer

12% compounded annually

Question 12

A company rents a booth at a fair for $500$ per month and sells muffins for $1.50$ each. If each muffin costs $0.40$ to make, what is the profit equation $\Pi$ in terms of $m$, the number of muffins sold?

Accepted Answer

$\Pi = 1.50m - 0.40m - 500$

Question 13

If $f(x) = 2x^2 - 2x$ and $g(x) = 3x - 1$, what is $g(f(2))$?

Accepted Answer

$g(4) = 3(4) - 1 = 11$

Question 14

A population is growing according to $P = 250(1.065)^t$. How many years will it take for the population to reach $1000$?

Accepted Answer

$t = \frac{\ln(4)}{\ln(1.065)} \approx 22.3$ years

Question 15

If $f(x) = 2x^2 - 2x$ and $g(x) = 3x - 1$, what is $f(g(x))$?

Accepted Answer

$f(g(x)) = 2(3x-1)^2 - 2(3x-1)$

Question 16

The half-life of cesium-137 is 30 years. How much of a 100-mg sample would remain after 50 years? (Round to the nearest mg.)

Accepted Answer

35 mg

Question 17

If $y = x^2 + x + 5$, what is the average rate of change of $y$ with respect to $x$ over the interval $3 \leq x \leq 6$?

Accepted Answer

$\frac{y(6) - y(3)}{6 - 3} = \frac{47 - 17}{3} = 10$

Question 18

If you invest your money in an account paying 4% interest per year, compounded annually, how long will it take for your money to double?

Accepted Answer

$t = \frac{\ln(2)}{\ln(1.04)} \approx 17.7$ years

Question 19

What interest rate, compounded continuously, is equivalent to a 7% rate compounded annually?

Accepted Answer

$r = \ln(1.07) \approx 0.0677 = 6.77\%$

Question 20

A business discovers a linear relationship between the number of shirts $S$ it can sell and the price per shirt $p$. In particular, 12,000 shirts can be sold at $\$19$ each, and 3,000 at $\$35$ each. What is the equation for $S$ as a function of $p$?

Accepted Answer

$S = -500p + 21,500$

Business Calculus Study Guide: Functions, Rates of Change, Exponential Models, and Applications

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