Q: A company’s cost function is $C(x) = 0.5x^2 + 20x + 100$. What is the average cost per item when $x = 10$?

Average cost is $\frac{C(10)}{10} = \frac{0.5 \times 100 + 20 \times 10 + 100}{10} = \frac{5 + 200 + 100}{10} = \frac{305}{10} = 30.5$

Question 1

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

Accepted Answer

A function that can be written as $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where $n$ is a non-negative integer and $a_n, \ldots, a_0$ are real numbers.

Question 2

A pharmaceutical company models the concentration of a drug in the bloodstream over time with the function $C(t) = 5e^{-0.2t}$, where $t$ is in hours. What type of function is this?

Accepted Answer

Exponential function

Question 3

Which of the following is the domain of the function $f(x) = \frac{1}{x-3}$?

Accepted Answer

All real numbers except $x = 3$

Question 4

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

Accepted Answer

Maximum profit is $P(10) = 100$ thousand dollars when $x = 10$

Question 5

Which of the following equations represents a circle with center $(2, -3)$ and radius $5$?

Accepted Answer

$(x-2)^2 + (y+3)^2 = 25$

Question 6

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

Accepted Answer

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

Question 7

Which of the following is the solution set for $|2x - 5| = 9$?

Accepted Answer

$x = 7$ or $x = -2$

Question 8

A car rental company charges a flat fee of $\$30$ plus $\$0.20$ per mile driven. Which equation models the total cost $C$ for driving $m$ miles?

Accepted Answer

$C = 30 + 0.20m$

Question 9

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

Accepted Answer

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

Question 10

A biologist uses the function $f(x) = \ln(x)$ to model the growth rate of a species. What is the domain of $f(x)$?

Accepted Answer

All real numbers $x > 0$

Question 11

Solve the system of equations: $\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$

Accepted Answer

$x = 2$, $y = 1$

Question 12

Which of the following is a rational function?

Accepted Answer

$f(x) = \frac{x^2 + 1}{x - 4}$

Question 13

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

Accepted Answer

$R(50) = 2500$

Question 14

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

Accepted Answer

All real numbers $y \geq -4$

Question 15

A scientist models the decay of a radioactive substance with $N(t) = N_0e^{-kt}$. If $N_0 = 100$ grams and $k = 0.05$, how much remains after $10$ years?

Accepted Answer

$N(10) = 100e^{-0.5} \approx 60.65$ grams

Question 16

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

Accepted Answer

$x = 2$ or $x = 4$

Question 17

A hospital uses the matrix $A = \begin{bmatrix} 2 & 3 \ 1 & 4 \end{bmatrix}$ to represent patient flow between departments. What is the determinant of $A$?

Accepted Answer

$2 	imes 4 - 1 	imes 3 = 5$

Question 18

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

Accepted Answer

$x = 32$

Question 19

A company’s cost function is $C(x) = 0.5x^2 + 20x + 100$. What is the average cost per item when $x = 10$?

Accepted Answer

Average cost is $\frac{C(10)}{10} = \frac{0.5 	imes 100 + 20 	imes 10 + 100}{10} = \frac{5 + 200 + 100}{10} = \frac{305}{10} = 30.5$

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