Q: A chemical reaction's rate $$R (in $$mol/L·s) is modeled by the exponential function $$R(t) = 5e$$^{-0.3t}$$, where t$ is time in seconds. What is the rate after 10 seconds, rounded to two decimal places?

$$R(10) = 5e$$^{-0.3 \times 10}$$ = 5e$$^{-3}$$ \approx 0.25$$

Question 1

A chemical reaction's rate $$R (in $$mol/L·s) is modeled by the exponential function $$R(t) = 5e$$^{-0.3t}$$, where t$ is time in seconds. What is the rate after 10 seconds, rounded to two decimal places?

Accepted Answer

$$R(10) = 5e$$^{-0.3 	imes 10}$$ = 5e$$^{-3}$$ \approx 0.25$$

Question 2

A company’s profit $$P (in $$thousands of dollars) is modeled by the quadratic function $$P(x) = -2x^2$ + 12x - 7$$, where $$x is $$the number of units (in hundreds) produced. What is the maximum profit, and how many units should be produced to achieve it?

Accepted Answer

Maximum profit is $$P(3) = 11$$ thousand dollars at $$x = 3 (i.e$$., 300 units)

Question 3

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

Accepted Answer

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

Question 4

Given the function $$f(x) = \frac{2x+1}{x-3}$$, what is the domain of $$f$$?

Accepted Answer

All real numbers except $$x = 3$$

Question 5

A biologist models the population of a bacteria culture by $$P(t) = 200 \cdot 2$$^{t}$$, where t$ is in hours. How many bacteria will there be after 5 hours?

Accepted Answer

$$P(5) = 200 \cdot 2$$^{5}$$ = 200 \cdot 32 = 6400$$

Question 6

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

Accepted Answer

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

Question 7

A line passes through the points $$(1, 2)$$ and $$(5, 10). $$What is the equation of the line in slope-intercept form?

Accepted Answer

$$y = 2x$$

Question 8

Solve the equation $$2x^2 - 8x + 6 = 0$ using the quadratic formula.

Accepted Answer

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

Question 9

Which of the following is equivalent to $$\log_{2} 32$$?

Accepted Answer

$$5$$

Question 10

A pharmaceutical company needs to mix two solutions: one with 10% concentration and one with 40% concentration to obtain 100 mL of a 25% solution. How many mL of the 10% solution should be used?

Accepted Answer

$$50 mL$$

Question 11

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

Accepted Answer

$$(f \circ g)(2) = f(g(2)) = f(3) = 9$$

Question 12

Which of the following is the solution set for the inequality $$|2x - 5| \u003c 7$$?

Accepted Answer

$$-1 \u003c x \u003c 6$$

Question 13

A circle has endpoints of a diameter at $$(2, 3)$$ and $$(8, 11). $$What is the center of the circle?

Accepted Answer

$$(5, 7)$$

Question 14

Which of the following is a solution to the system of equations: $$\begin{cases} 2x + y = 7 \ x - y = 1 \end{cases}$$?

Accepted Answer

$$x = 2$$, $$y = 3$$

Question 15

Which of the following is the correct application of the Remainder Theorem for $$f(x) = x^3$ - 4x^2$ + 5x - 2$$ when divided by $$x - 2$$?

Accepted Answer

The remainder is $$f(2) = 8 - 16 + 10 - 2 = 0$$

Question 16

Which of the following is the correct simplification of $$\log_{3} 81$$?

Accepted Answer

$$4$$

Question 17

A car depreciates in value according to the function $$V(t) = 20000 \cdot (0.85)^t$$, where $$t is $$the number of years after purchase. What is the value of the car after 3 years?

Accepted Answer

$$V(3) = 20000 \cdot (0.85)^3$ \approx 12262.5$$

Question 18

Which of the following is the correct solution to the equation $$3^{x} = 81$?

Accepted Answer

$$x = 4$$

Question 19

If $$f(x) = 2x^2$ - 3x + 1$$, what is $$f(-2)$$?

Accepted Answer

$$f(-2) = 2(-2)^2$ - 3(-2) + 1 = 8 + 6 + 1 = 15$$

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