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 $$x = 100 to$$ $$x = 150$$?

Accepted Answer

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

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