Q: A hospital uses a mathematical model to predict the spread of a virus. The number of cases $$N$$ after $$t$$ days is given by $$N(t) = 100e$$^{0.05t}$$. How many cases will there be after 20$ days?

$$N(20) = 100e$$^{0.05 \times 20}$$ = 100e$$^{1}$$ \approx 271.83$$

Q: A pharmaceutical company models the concentration $$C of a $$drug in the bloodstream over time $$t (in $$hours) as $$C(t) = 50e$$^{-0.2t}$$. What is the concentration after 5$ hours?

$$C(5) = 50e$$^{-0.2 \times 5}$$ = 50e$$^{-1}$$ \approx 18.39$$

Q: A population of bacteria doubles every $$3$$ hours. If the initial population is $$500$$, which equation models the population $$P$$ after $$t$$ hours?

$$P(t) = 500 \times 2$$^{t/3}$$

Q: A chemical reaction follows the equation $$y = 4x^2$ - 3x + 2$$, where $$y is $$the concentration and $$x is $$time in hours. What is the concentration after $$x = 2$$ hours?

$$y = 4 \times 2^2$ - 3 \times 2 + 2 = 16 - 6 + 2 = 12$$

Q: A city’s population is modeled by $$P(t) = 2000 \times (1.03)^t$$, where $$t is $$years since 2020. What will the population be in 2025?

$$P(5) = 2000 \times (1.03)^5$ \approx 2000 \times 1.159274 \approx 2318.55$$

Question 1

Which of the following best describes the unit circle in the context of trigonometry?

Accepted Answer

A circle with radius $$1$$ centered at $$(0,0)$$

Question 2

A hospital uses a mathematical model to predict the spread of a virus. The number of cases $$N$$ after $$t$$ days is given by $$N(t) = 100e$$^{0.05t}$$. How many cases will there be after 20$ days?

Accepted Answer

$$N(20) = 100e$$^{0.05 	imes 20}$$ = 100e$$^{1}$$ \approx 271.83$$

Question 3

Which of the following is the correct definition of a function?

Accepted Answer

A relation where each input is paired with exactly one output.

Question 4

A pharmaceutical company models the concentration $$C of a $$drug in the bloodstream over time $$t (in $$hours) as $$C(t) = 50e$$^{-0.2t}$$. What is the concentration after 5$ hours?

Accepted Answer

$$C(5) = 50e$$^{-0.2 	imes 5}$$ = 50e$$^{-1}$$ \approx 18.39$$

Question 5

Which of the following is the solution to the equation $$2x^2 - 8x + 6 = 0$?

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 6

A population of bacteria doubles every $$3$$ hours. If the initial population is $$500$$, which equation models the population $$P$$ after $$t$$ hours?

Accepted Answer

$$P(t) = 500 	imes 2$$^{t/3}$$

Question 7

Which of the following is the inverse function of $$f(x) = 3x + 7$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x - 7}{3}$$

Question 8

A car rental company charges a flat fee of $$\25$ plus $$\$$0.15$$ per mile driven. Which function models the total cost $$C$$ for $$m$$ miles?

Accepted Answer

$$C(m) = 25 + 0.15m$$

Question 9

Which of the following is the correct solution to the system of equations: $$\begin{cases} x + y = 5 \ 2x - y = 1 \end{cases}$$?

Accepted Answer

$$x = 2$$, $$y = 3$$

Question 10

A chemical reaction follows the equation $$y = 4x^2$ - 3x + 2$$, where $$y is $$the concentration and $$x is $$time in hours. What is the concentration after $$x = 2$$ hours?

Accepted Answer

$$y = 4 	imes 2^2$ - 3 	imes 2 + 2 = 16 - 6 + 2 = 12$$

Question 11

Which of the following is the correct domain of the function $$f(x) = \frac{1}{x-4}$$?

Accepted Answer

All real numbers except $$x = 4$$

Question 12

A company’s profit $$P is $$modeled by $$P(x) = -2x^2$ + 12x - 20$$, where $$x is $$the number of units sold. What is the maximum profit, and for how many units?

Accepted Answer

Maximum at $$x = 3$$, $$P(3) = 10$$

Question 13

Which of the following is the correct range of the function $$f(x) = x^2$?

Accepted Answer

All real numbers $$y \geq 0$$

Question 14

A city’s population is modeled by $$P(t) = 2000 	imes (1.03)^t$$, where $$t is $$years since 2020. What will the population be in 2025?

Accepted Answer

$$P(5) = 2000 	imes (1.03)^5$ \approx 2000 	imes 1.159274 \approx 2318.55$$

Question 15

Which of the following is the correct solution to $$\log_2$(x) = 5$$?

Accepted Answer

$$x = 32$$

Question 16

A medical researcher uses the function $$f(x) = \sqrt{x-1} to $$model a response. What is the domain of $$f$$?

Accepted Answer

$$x \geq 1$$

Question 17

Which of the following is the correct solution to the equation $$|2x - 5| = 7$$?

Accepted Answer

$$x = 6 or$$ $$x = -1$$

Question 18

A company’s revenue is modeled by $$R(x) = 100x - 0.5x^2$, where x$ is the number of items sold. What is the maximum revenue?

Accepted Answer

Maximum at $$x = 100$$, $$R(100) = 100 	imes 100 - 0.5 	imes 100^2$ = 10000 - 5000 = 5000$$

Question 19

Which of the following is the correct solution to the quadratic equation $$x^2 - 6x + 9 = 0$?

Accepted Answer

$$x = 3$$

College Algebra & Precalculus: Assignment and Assessment Structure Overview

스터디 가이드 - 연습 문제

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