Q: A company’s profit $$P is $$modeled by the polynomial $$P(x) = x$$^{3}$$ - 4x$$^{2}$$ + 5x - 2. $$What is the profit when $$x = 2$$?

$$P(2) = 2$$^{3}$$ - 4 \times 2$$^{2}$$ + 5 \times 2 - 2 = 8 - 16 + 10 - 2 = 0$$

Q: A scientist records the temperature $$T in $$degrees Celsius as a function of time $$t in $$hours, modeled by $$T(t) = 20 + 5t - t$$^{2}$$. What is the temperature after 3 hours?

$$T(3) = 20 + 5 \times 3 - 3$$^{2}$$ = 20 + 15 - 9 = 26$$ degrees Celsius

Q: A patient’s heart rate $$H is $$modeled by $$H = 70 + 10\sin(t)$$, where $$t is $$time in minutes. What is the heart rate at $$t = \dfrac{\pi}{2}$$ minutes?

$$H = 70 + 10 \sin\left(\dfrac{\pi}{2}\right) = 70 + 10 \times 1 = 80$$

Question 1

A biologist is modeling the population growth of bacteria using the function $$P(t) = 500 \cdot 2$$^{t}$$, where t$ is the number of hours. What is the population after 5 hours?

Accepted Answer

$$P(5) = 500 \cdot 2$$^{5}$$ = 16,000$$

Question 2

Which of the following is the correct simplification of the expression $$\sqrt[3]{x$$^{9}$$y^{6}$$}$?

Accepted Answer

$$x^{3}y$$^{2}$

Question 3

A chemical reaction rate $$R is $$inversely proportional to the concentration $$C of a $$reactant. Which equation models this relationship?

Accepted Answer

$$R = \dfrac{k}{C}$$

Question 4

Which of the following is the correct factorization of the polynomial $$x^{2} - 5x + 6$?

Accepted Answer

$$(x-2)(x-3)$$

Question 5

A medical dosage formula is given by $$D = \dfrac{mg}{kg}$$, where $$mg is $$the milligrams of drug and $$kg is $$the patient's weight. If a patient weighs 70 kg and needs a dosage of 2 mg/kg, what is the total dosage?

Accepted Answer

$$2 	imes 70 = 140 mg$$

Question 6

Simplify the rational expression $$\dfrac{x$$^{2}$$ - 9}{x$$^{2}$$ - 6x + 9}.$$

Accepted Answer

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

Question 7

Which of the following is the correct solution to the quadratic equation $$x^{2} + 4x + 3 = 0$?

Accepted Answer

$$x = -1$$ and $$x = -3$$

Question 8

A company’s profit $$P is $$modeled by the polynomial $$P(x) = x$$^{3}$$ - 4x$$^{2}$$ + 5x - 2. $$What is the profit when $$x = 2$$?

Accepted Answer

$$P(2) = 2$$^{3}$$ - 4 	imes 2$$^{2}$$ + 5 	imes 2 - 2 = 8 - 16 + 10 - 2 = 0$$

Question 9

Which of the following is the correct simplification of $$\dfrac{2x$$^{2}$$y}{4xy$$^{2}$$}$$?

Accepted Answer

$$\dfrac{x}{2y}$$

Question 10

A scientist records the temperature $$T in $$degrees Celsius as a function of time $$t in $$hours, modeled by $$T(t) = 20 + 5t - t$$^{2}$$. What is the temperature after 3 hours?

Accepted Answer

$$T(3) = 20 + 5 	imes 3 - 3$$^{2}$$ = 20 + 15 - 9 = 26$$ degrees Celsius

Question 11

Which of the following is the correct solution to the equation $$\dfrac{2}{x} = 4$$?

Accepted Answer

$$x = \dfrac{2}{4} = 0.5$$

Question 12

Which of the following is the correct simplification of $$\$$left(3x^{2}$$y\$$right)^{3}$$?

Accepted Answer

$$27x^{6}y$$^{3}$

Question 13

A patient’s heart rate $$H is $$modeled by $$H = 70 + 10\sin(t)$$, where $$t is $$time in minutes. What is the heart rate at $$t = \dfrac{\pi}{2}$$ minutes?

Accepted Answer

$$H = 70 + 10 \sin\left(\dfrac{\pi}{2}ight) = 70 + 10 	imes 1 = 80$$

Question 14

Which of the following is the correct simplification of $$\dfrac{5x$$^{2}$$y^{3}$$}{10xy}$?

Accepted Answer

$$\dfrac{x y$$^{2}$$}{2}$$

Question 15

Which of the following is the correct solution to the equation $$x^{2} = -16$ in the set of complex numbers?

Accepted Answer

$$x = 4i$$ and $$x = -4i$$

Question 16

A car’s value $$V$$ depreciates according to $$V = 20,000 \cdot 0.85$$^{t}$$, where t$ is the number of years. What is the value after 3 years?

Accepted Answer

$$V = 20,000 \cdot 0.85$$^{3}$$ \approx 12,281.25$$

Question 17

Which of the following is the correct simplification of $$\dfrac{1}{\sqrt{5}}$$?

Accepted Answer

$$\dfrac{\sqrt{5}}{5}$$

Question 18

Which of the following is the correct solution to the equation $$2x^{2} - 8x = 0$?

Accepted Answer

$$x = 0$$ and $$x = 4$$

Question 19

A graphing utility plots the function $$f(x) = x$$^{2}$$ - 4x + 7. $$What is the vertex of the parabola?

Accepted Answer

$$(2, 3)$$

Precalculus Foundations: Algebraic Expressions, Equations, and Graphs

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