Q: A chemist mixes two solutions, one with a concentration of $30\%$ and another with $50\%$ acid, to obtain $10$ liters of a $40\%$ acid solution. Let $x$ be the amount (in liters) of the $30\%$ solution used. Which equation models this scenario?

$0.3x + 0.5(10 - x) = 0.4 \times 10$

Question 1

A pharmaceutical company is developing a new drug and models the concentration of the drug in the bloodstream over time with the polynomial $C(t) = -0.5t^3 + 2t^2 + 3t + 10$, where $t$ is the time in hours. What is the degree and leading coefficient of this polynomial?

Accepted Answer

Degree: $3$, Leading coefficient: $-0.5$

Question 2

Which of the following expressions is NOT a polynomial?

Accepted Answer

$x^{-2} + 5$

Question 3

A rectangular garden has a length that is 3 meters longer than twice its width. If the area of the garden is $54$ square meters, which equation models the width $w$ of the garden?

Accepted Answer

$2w^2 + 3w - 54 = 0$

Question 4

Factor the polynomial $x^2 - 2x - 15$ completely.

Accepted Answer

$(x - 5)(x + 3)$

Question 5

A biologist is studying the population of a species, modeled by $P(t) = 1000e^{0.03t}$, where $t$ is in years. If the population doubles, what is the value of $t$? (Round to the nearest year.)

Accepted Answer

$t \approx 23$

Question 6

Simplify the following expression: $\dfrac{x^2 + 2x}{4x}$

Question 7

Solve for $x$: $2x - 5 = 9$

Accepted Answer

$x = 7$

Question 8

A chemist mixes two solutions, one with a concentration of $30\%$ and another with $50\%$ acid, to obtain $10$ liters of a $40\%$ acid solution. Let $x$ be the amount (in liters) of the $30\%$ solution used. Which equation models this scenario?

Accepted Answer

$0.3x + 0.5(10 - x) = 0.4 	imes 10$

Question 9

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

Accepted Answer

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

Question 10

A ball is thrown upward from the ground with an initial velocity of $32$ ft/s. Its height after $t$ seconds is given by $h(t) = -16t^2 + 32t$. At what time does the ball return to the ground?

Accepted Answer

$t = 2$

Question 11

Solve the equation $x^2 - 4x + 3 = 0$ for $x$.

Accepted Answer

$x = 1, 3$

Question 12

Which of the following is the solution set for $|2x - 6| = 8$?

Accepted Answer

$x = 7, x = -1$

Question 13

A company’s profit $P$ (in thousands of dollars) is modeled by $P(x) = -2x^2 + 8x + 10$, where $x$ is the number of units sold (in thousands). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit: $18$ at $x = 2$

Question 14

Simplify the expression $\sqrt{50} + 2\sqrt{8}$.

Accepted Answer

$7\sqrt{2}$

Question 15

Solve for $x$: $\dfrac{3}{x} = 2$

Accepted Answer

$x = 1.5$

Question 16

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

Accepted Answer

$x > 7$

Question 17

A medical researcher is analyzing the decay of a radioactive isotope, modeled by $A(t) = A_0 e^{-0.1t}$, where $A_0$ is the initial amount and $t$ is in years. What is the half-life of the isotope? (Round to the nearest year.)

Accepted Answer

$t \approx 7$

Question 18

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

Accepted Answer

$x = -1 \pm i$

Question 19

A car rental company charges a flat fee of $\$20$ plus $\$0.15$ per mile driven. If a customer paid $\$50$, how many miles did they drive?

Accepted Answer

$200$ miles

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