Q: The population of a country is modeled by $$P(t) = 4e$$^{0.11t}$$, where t$ is years since 2000. In what year will the population reach 12 million?

$$t = \frac{\ln(3)}{0.11}$$,\ \text{so year 2015}$

Question 1

Which of the following best describes the domain of the function $$f(x) = \ln(6 - x)$$?

Accepted Answer

$$x \in (-\infty, 6)$$

Question 2

Given the function $$f(x) = 5e^x + 1$$ for $$x \u003e 1$$, what is the inverse function $$f^{-1}(y$$)$$?

Accepted Answer

$$f^{-1}(y$$) = \ln\left(\frac{y-1}{5}\right)$$

Question 3

A rectangular storage container with an open top has a volume of $$10\ \mathrm{m}^3. $$The length of its base is twice its width. Material for the base costs $$10 per square meter, and material for the sides costs 6 $$per square meter. If $$w is $$the width of the base, which expression gives the cost $$C(w) of $$materials as a function of $$w$$?

Accepted Answer

$$C(w) = 20w^2$ + 60\left(\frac{10}{2w^2$}\right)$$

Question 4

Solve for $$x$$: $$2\ln(3x) + 5 = 7.$$

Accepted Answer

$$x = \frac{e}{3}$$

Question 5

The population of a country is modeled by $$P(t) = 4e$$^{0.11t}$$, where t$ is years since 2000. In what year will the population reach 12 million?

Accepted Answer

$$t = \frac{\ln(3)}{0.11}$$,\ 	ext{so year 2015}$

Question 6

A company has $$C = g(t)$$ thousand dollars in its cash reserve $$t$$ years after January 1, 2023. Every year this company increases its cash reserves by $$40$$ thousand dollars, and on January 1, 2025 it will have $$90$$ thousand dollars in reserve. What was the reserve on January 1, 2023?

Accepted Answer

$$50$$ thousand dollars

Question 7

Which of the following best describes the function $$C = h(t) if a $$company starts with $$40$$ thousand dollars and increases its cash reserves by $$40\%$$ each year?

Accepted Answer

$$C = 40(1.4)^t$$

Question 8

What is the period of the function $$f(x) = 3 + 4\sin(4x)$$?

Accepted Answer

$$\pi$$

Question 9

A student club sells T-shirts for $$10 each and sells 500 shirts. For every 1$ price increase, they sell 10$ fewer shirts. What is the demand function N(P$$)$$, where P$ is the price?

Accepted Answer

$$N(P) = 500 - 10(P - 10)$$

Question 10

Given the function $$f(x) = \sqrt{16 - 2x}$$, what is its domain?

Accepted Answer

$$(-\infty, 8]$$

Question 11

Solve for $$x$$: $$5e^{3x+2} = 20$.

Accepted Answer

$$x = \frac{\ln 4 - 2}{3}$$

Question 12

Which equation best represents the statement: 'After 25 minutes online, the website has earned $457' dollars'?

Accepted Answer

$$f(25) = 457$$

Question 13

The number of daylight hours $$D(t) in a $$city oscillates between $$9.5$$ and $$14.5$$ hours per year, with the longest day $$170$$ days after January 1. Which formula models $$D(t)$$?

Accepted Answer

$$D(t) = 12 + 2.5\sin\left(\frac{2\pi}{364}(t - 170)\right)$$

Question 14

Find all solutions to $$5\sin(3x + 2) = 3.$$

Accepted Answer

$$3x + 2 = \arcsin\left(\frac{3}{5}\right) + 2\pi n or$$ $$3x + 2 = \pi - \arcsin\left(\frac{3}{5}\right) + 2\pi n$$

Question 15

Which of the following is the correct domain for $$f(x) = \ln(5x + 2) - 3$$?

Accepted Answer

$$x \in \left( -\frac{2}{5}, \infty \right)$$

Question 16

If an investment increases by $$12\%$$ per year, how long does it take for the value to increase by a factor of $$5$$?

Accepted Answer

$$t = \frac{\ln 5}{\ln 1.12}$$

Question 17

Which of the following best describes the range of the function $$f(x) = \sqrt{16 - 2x}$$?

Accepted Answer

$$[0, 4]$$

Question 18

Which of the following best describes the function modeling cash reserves that oscillate yearly between $$50$$ and $$250$$ thousand dollars, with a minimum at $$t=0$$ months?

Accepted Answer

$$C(t) = 150 + 100\cos\left(\frac{2\pi}{12}t + \pi\right)$$

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