Q: A biologist models the population of a bacteria culture with $$P(t) = 500e$$^{0.3t}$$, where t$ is time in hours. What is the population after 5 hours?

$$P(5) = 500e$$^{0.3 \times 5}$$ = 500e$$^{1.5}$$ \approx 500 \times 4.4817 \approx 2240.85$$

Q: A company’s profit $$P (in $$dollars) is modeled by $$P(x) = -2x^2$ + 120x - 1000$$, where $$x is $$the number of units sold. How many units must be sold to maximize profit?

$$x = \dfrac{-b}{2a} = \dfrac{-120}{2 \times -2} = \dfrac{-120}{-4} = 30$$

Q: A medical researcher uses the exponential decay model $$N(t) = 1000e$$^{-0.2t}$$ to describe the amount of a drug in the bloodstream after t$ hours. How much drug remains after 8 hours?

$$N(8) = 1000e$$^{-0.2 \times 8}$$ = 1000e$$^{-1.6}$$ \approx 1000 \times 0.2019 \approx 201.9$$

Q: A car’s value $$V$$ depreciates according to $$V(t) = 20000(0.85)^t$$, where $$t is $$the number of years after purchase. What is the value after 3 years?

$$V(3) = 20000(0.85)^3$ = 20000 \times 0.614125 = 12282.50$$

Q: A chemical reaction is modeled by $$C(t) = 100(1 - e$$^{-0.4t}$$)$$, where $$C is $$the concentration after $$t$$ minutes. What is the concentration after 10 minutes?

$$C(10) = 100(1 - e$$^{-0.4 \times 10}$$) = 100(1 - e$$^{-4}$$) \approx 100(1 - 0.0183) = 98.17$$

Question 1

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

Accepted Answer

A relation in which each input is paired with exactly one output.

Question 2

Given the function $$f(x) = 2x^2$ - 3x + 5$$, what is $$f(-2)$$?

Accepted Answer

$$f(-2) = 2(-2)^2$ - 3(-2) + 5 = 8 + 6 + 5 = 19$$

Question 3

Which of the following equations represents a linear function?

Accepted Answer

$$y = 4x - 7$$

Question 4

A biologist models the population of a bacteria culture with $$P(t) = 500e$$^{0.3t}$$, where t$ is time in hours. What is the population after 5 hours?

Accepted Answer

$$P(5) = 500e$$^{0.3 	imes 5}$$ = 500e$$^{1.5}$$ \approx 500 	imes 4.4817 \approx 2240.85$$

Question 5

Which of the following is the solution set for the inequality $$2x - 5 \u003e 9$$?

Accepted Answer

$$x \u003e 7$$

Question 6

A company’s profit $$P (in $$dollars) is modeled by $$P(x) = -2x^2$ + 120x - 1000$$, where $$x is $$the number of units sold. How many units must be sold to maximize profit?

Accepted Answer

$$x = \dfrac{-b}{2a} = \dfrac{-120}{2 	imes -2} = \dfrac{-120}{-4} = 30$$

Question 7

Which of the following is the correct solution to the system of equations: $$\begin{cases} 2x + y = 7 \ x - y = 1 \end{cases}$$?

Accepted Answer

$$(x, y) = (2, 3)$$

Question 8

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

Accepted Answer

$$x \in \mathbb{R}, x 
eq 4$$

Question 9

A medical researcher uses the exponential decay model $$N(t) = 1000e$$^{-0.2t}$$ to describe the amount of a drug in the bloodstream after t$ hours. How much drug remains after 8 hours?

Accepted Answer

$$N(8) = 1000e$$^{-0.2 	imes 8}$$ = 1000e$$^{-1.6}$$ \approx 1000 	imes 0.2019 \approx 201.9$$

Question 10

Which of the following is the correct factorization of $$x^2 - 9$?

Accepted Answer

$$(x - 3)(x + 3)$$

Question 11

A car’s value $$V$$ depreciates according to $$V(t) = 20000(0.85)^t$$, where $$t is $$the number of years after purchase. What is the value after 3 years?

Accepted Answer

$$V(3) = 20000(0.85)^3$ = 20000 	imes 0.614125 = 12282.50$$

Question 12

Which of the following is the solution to $$|3x - 7| = 5$$?

Accepted Answer

$$x = 4 or$$ $$x = \dfrac{2}{3}$$

Question 13

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

Accepted Answer

$$x = 2 or$$ $$x = 4$$

Question 14

A student invests $$\500$ in an account that earns 6$$\%$$ annual interest compounded monthly. What is the value after 2 years? Use A = P$$\left(1 + \dfrac{r}{n}\$$right)^{nt}$.

Accepted Answer

$$A = 500\left(1 + \dfrac{0.06}{12}\$$right)^{24}$$ \approx 500 	imes 1.1275 = 563.75$$

Question 15

Which of the following is the inverse function of $$f(x) = 3x + 2$$?

Accepted Answer

$$f^{-1}(x$$) = \dfrac{x - 2}{3}$$

Question 16

Which of the following is the correct solution to the equation $$\log_{2}(x) = 5$$?

Accepted Answer

$$x = 32$$

Question 17

A chemical reaction is modeled by $$C(t) = 100(1 - e$$^{-0.4t}$$)$$, where $$C is $$the concentration after $$t$$ minutes. What is the concentration after 10 minutes?

Accepted Answer

$$C(10) = 100(1 - e$$^{-0.4 	imes 10}$$) = 100(1 - e$$^{-4}$$) \approx 100(1 - 0.0183) = 98.17$$

Question 18

Which of the following is the correct solution to the equation $$2^{x} = 20$?

Accepted Answer

$$x = \log_{2}(20) \approx 4.32$$

Question 19

Which of the following is the range of the function $$f(x) = x^2$ - 4$$?

Accepted Answer

$$y \geq -4$$

College Algebra Syllabus and Course Overview

Guida di studio - Domande di pratica

Study guide - Flashcard