Q: A chemical reaction follows the equation $A = A_0e^{-kt}$, where $A_0 = 100$ mg, $k = 0.1$, and $t = 10$ hours. What is the amount of substance $A$ left after 10 hours?

$A = 100e^{-1} \approx 100 \cdot 0.3679 \approx 36.8$ mg

Q: A radioactive isotope has a half-life of 6 hours. If the initial amount is 200 mg, how much remains after 18 hours? Use $A = A_0 \left(\frac{1}{2}\right)^{t/6}$.

$A = 200 \left(\frac{1}{2}\right)^{3} = 200 \cdot \frac{1}{8} = 25$ mg

Q: A business’s profit $P$ is given by $P(x) = -2x^2 + 40x - 100$, where $x$ is the number of items sold. What is the maximum profit, and how many items must be sold to achieve it?

Maximum profit at $x = 10$, $P(10) = -2 \cdot 100 + 40 \cdot 10 - 100 = -200 + 400 - 100 = 100$

Question 1

A biologist is tracking the population of a bacteria culture, which doubles every 3 hours. If the initial population is 500, what will the population be after 12 hours? Use the exponential growth formula $P(t) = P_0 \cdot 2^{t/3}$, where $t$ is in hours.

Accepted Answer

$P(12) = 500 \cdot 2^{4} = 500 \cdot 16 = 8000$

Question 2

Find the distance between the points $A(2, -3)$ and $B(-4, 5)$ using the distance formula.

Accepted Answer

$\sqrt{(2 + 4)^2 + (-3 - 5)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

Question 3

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

Accepted Answer

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

Question 4

A company’s revenue $R$ (in thousands of dollars) is modeled by $R(x) = 5x^2 + 20x + 100$, where $x$ is the number of units sold (in thousands). What is the revenue if 3,000 units are sold?

Accepted Answer

$R(3) = 5 \cdot 3^2 + 20 \cdot 3 + 100 = 5 \cdot 9 + 60 + 100 = 45 + 60 + 100 = 205$

Question 5

Which of the following is the correct composition of the functions $f(x) = x^2$ and $g(x) = x + 1$ evaluated at $x = 2$ for $(f \circ g)(x)$?

Accepted Answer

$(f \circ g)(2) = f(g(2)) = f(3) = 3^2 = 9$

Question 6

Solve the quadratic equation $x^2 - 5x + 6 = 0$.

Accepted Answer

$x = 2$ and $x = 3$

Question 7

A medical researcher models the decay of a drug in the bloodstream by $C(t) = 50e^{-0.2t}$, where $C$ is the concentration in mg/L and $t$ is time in hours. What is the concentration after 5 hours?

Accepted Answer

$C(5) = 50e^{-1} \approx 50 \cdot 0.3679 \approx 18.4$ mg/L

Question 8

Which of the following is the correct solution to the inequality $2x - 7 < 5$?

Accepted Answer

$x < 6$

Question 9

Given the function $f(x) = \frac{1}{x - 2}$, what is the domain of $f$?

Accepted Answer

$x \in \mathbb{R}, x 
eq 2$

Question 10

Expand the logarithmic expression $\ln(2x^3)$ using logarithm properties.

Accepted Answer

$\ln(2x^3) = \ln(2) + 3\ln(x)$

Question 11

A chemical reaction follows the equation $A = A_0e^{-kt}$, where $A_0 = 100$ mg, $k = 0.1$, and $t = 10$ hours. What is the amount of substance $A$ left after 10 hours?

Accepted Answer

$A = 100e^{-1} \approx 100 \cdot 0.3679 \approx 36.8$ mg

Question 12

Solve the system of equations: $\begin{cases} x + y = 7 \ x - y = 3 \end{cases}$

Accepted Answer

$x = 5, y = 2$

Question 13

Which of the following is the correct equation of a line passing through the points $(1, 2)$ and $(4, 8)$?

Accepted Answer

$y = 2x$

Question 14

A radioactive isotope has a half-life of 6 hours. If the initial amount is 200 mg, how much remains after 18 hours? Use $A = A_0 \left(\frac{1}{2}\right)^{t/6}$.

Accepted Answer

$A = 200 \left(\frac{1}{2}\right)^{3} = 200 \cdot \frac{1}{8} = 25$ mg

Question 15

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

Accepted Answer

$x = 32$

Question 16

Find the range of the function $f(x) = \sqrt{x - 4}$.

Accepted Answer

$[0, \infty)$

Question 17

A business’s profit $P$ is given by $P(x) = -2x^2 + 40x - 100$, where $x$ is the number of items sold. What is the maximum profit, and how many items must be sold to achieve it?

Accepted Answer

Maximum profit at $x = 10$, $P(10) = -2 \cdot 100 + 40 \cdot 10 - 100 = -200 + 400 - 100 = 100$

Question 18

Solve the system of equations for the intersection point: $\begin{cases} y = 2x + 1 \ y = -x + 4 \end{cases}$

Accepted Answer

$x = 1, y = 3$

Question 19

Which of the following is the correct formula for the inverse of $f(x) = \frac{2x - 5}{3}$?

Accepted Answer

$f^{-1}(x) = \frac{3x + 5}{2}$

Math 142 Final Exam Review: College Algebra Study Guide

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