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: $$\dfrac{x^2$ + 2x}{4x}$$

Accepted Answer

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 $$2x - 5 \u003e 9$$?

Accepted Answer

$$x \u003e 7$$

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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