Q: A pharmaceutical company models the concentration $$C (in mg/L) of a $$drug in a patient's bloodstream $$t$$ hours after administration by the function $$C(t) = 20e$$^{-0.3t}$$. What is the concentration after 4 hours? (Round your answer to two decimal places.)

$$C(4) = 20e$$^{-0.3 \times 4}$$ = 20e$$^{-1.2}$$ \approx 6.02 mg/L$$

Q: A biologist is studying the growth of a bacteria population, which doubles every 3 hours. If the initial population is 500, which function models the population $$P$$ after $$t$$ hours?

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

Q: If $$f(x) = 2x + 1$$ and $$g(x) = x^2$, what is (f$$ \circ g)(3)$$?

$$(f \circ g)(3) = f(g(3)) = f(9) = 2 \times 9 + 1 = 19$$

Q: A car’s value depreciates exponentially according to $$V(t) = 25000(0.85)^t$$, where $$t is $$the number of years after purchase. What is the value of the car after 5 years? (Round to the nearest dollar.)

$$V(5) = 25000 \times (0.85)^5$ \approx 12544$$

Question 1

A pharmaceutical company models the concentration $$C (in mg/L) of a $$drug in a patient's bloodstream $$t$$ hours after administration by the function $$C(t) = 20e$$^{-0.3t}$$. What is the concentration after 4 hours? (Round your answer to two decimal places.)

Accepted Answer

$$C(4) = 20e$$^{-0.3 	imes 4}$$ = 20e$$^{-1.2}$$ \approx 6.02 mg/L$$

Question 2

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

Accepted Answer

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

Question 3

A company’s profit $$P (in $$thousands of dollars) is modeled by the quadratic function $$P(x) = -2x^2$ + 12x - 16$$, where $$x is $$the number of units (in thousands) produced. What is the maximum profit, and for how many units produced does it occur?

Accepted Answer

Maximum profit is $$P(3) = -2(3)^2$ + 12(3) - 16 = 2$$ thousand dollars at $$x = 3$$ thousand units.

Question 4

Which of the following is the solution set for the equation $$2x^2 - 8x = 0$?

Accepted Answer

$$x = 0 or$$ $$x = 4$$

Question 5

A biologist is studying the growth of a bacteria population, which doubles every 3 hours. If the initial population is 500, which function models the population $$P$$ after $$t$$ hours?

Accepted Answer

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

Question 6

Which of the following is the correct simplification of the rational expression $$\frac{x^2$ - 9}{x^2$ - 6x + 9}$$?

Accepted Answer

$$\frac{(x - 3)(x + 3)}{(x - 3)^2$} = \frac{x + 3}{x - 3}$$, \quad x 
eq 3$

Question 7

Given the system of equations: $$\begin{cases} 2x + 3y = 7 \ 4x - y = 5 \end{cases}$$, what is the solution $$(x, y)$$?

Accepted Answer

$$(x, y) = (2, 1)$$

Question 8

Which of the following is the correct graph of the function $$f(x) = -x^2$ + 4x - 3$$?

Accepted Answer

A downward-opening parabola with vertex at $$(2, 1)$$ and $$y-$$intercept at $$-3$$

Question 9

A radioactive substance decays according to the formula $$A(t) = A_0$ e$$^{-kt}$$, where A_0$ is the initial amount and k$ is a positive constant. If A_0$ = 100$$ grams and $$k = 0.05$$, how much remains after 10 years? (Round to the nearest gram.)

Accepted Answer

$$A(10) = 100e$$^{-0.5}$$ \approx 61$$ grams

Question 10

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

Accepted Answer

$$x = 32$$

Question 11

A rectangular garden has a length that is 3 meters longer than its width. If the area is 70 $$m^2$$, what are the dimensions of the garden?

Accepted Answer

Width: $$7 m$$, Length: $$10 m$$

Question 12

Which of the following is the correct result of dividing $$\frac{6x^3$ - 9x^2$}{3x}$$?

Accepted Answer

$$2x^2 - 3x$

Question 13

If $$f(x) = 2x + 1$$ and $$g(x) = x^2$, what is (f$$ \circ g)(3)$$?

Accepted Answer

$$(f \circ g)(3) = f(g(3)) = f(9) = 2 	imes 9 + 1 = 19$$

Question 14

Which of the following is the correct solution to the equation $$\sqrt{x + 5} = 4$$?

Accepted Answer

$$x = 11$$

Question 15

A car’s value depreciates exponentially according to $$V(t) = 25000(0.85)^t$$, where $$t is $$the number of years after purchase. What is the value of the car after 5 years? (Round to the nearest dollar.)

Accepted Answer

$$V(5) = 25000 	imes (0.85)^5$ \approx 12544$$

Question 16

Which of the following is the correct solution to the system of equations using matrices: $$\begin{cases} x + 2y = 8 \ 3x - y = 7 \end{cases}$$?

Accepted Answer

$$(x, y) = (3, 2.5)$$

Question 17

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

Accepted Answer

$$\frac{x}{2y}$$

Question 18

If $$f(x) = x^2$ - 4x + 7$$, what is the minimum value of $$f(x)$$?

Accepted Answer

The minimum value is $$f(2) = 3$$

Question 19

Which of the following is the correct solution to the equation $$2^{x} = 16$?

Accepted Answer

$$x = 4$$

Foundation of College Algebra: Mini-Textbook Study Notes

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