Q: A population of bacteria doubles every 3 hours. If the initial population is 200, which function models the population $$P(t)$$ after $$t$$ hours?

$$P(t) = 200 \times 2$$^{t/3}$$

Question 1

A chemical reaction's rate is modeled by the exponential function $$R(t) = 500e$$^{-0.2t}$$, where t$ is time in hours. What is the rate after 5 hours?

Accepted Answer

$$R(5) = 500e$$^{-1}$$ \approx 184$$

Question 2

Which of the following is the correct formula for the Law of Cosines used to determine the length of side $$a in a $$triangle with sides $$a$$, $$b$$, $$c$$ and angle $$A$$ opposite side $$a$$?

Accepted Answer

$$a^2$$ = $$b^2$$ + $$c^2 - 2bc$$\cos A$$

Question 3

A population of bacteria doubles every 3 hours. If the initial population is 200, which function models the population $$P(t)$$ after $$t$$ hours?

Accepted Answer

$$P(t) = 200 	imes 2$$^{t/3}$$

Question 4

Which of the following is the correct identity for $$\sin(2x)$$?

Accepted Answer

$$\sin(2x) = 2\sin x \cos x$$

Question 5

A medical dosage is modeled by the function $$D(t) = 100e$$^{-0.5t}$$, where t$ is in hours. How long will it take for the dosage to decrease to 25 mg?

Accepted Answer

$$t = 2.77$$ hours

Question 6

Which of the following is the correct expression for the sum formula for sine, $$\sin(x+y)$$?

Accepted Answer

$$\sin(x+y) = \sin x \cos y + \cos x \sin y$$

Question 7

A parabola models the path of a projectile: $$y = -2x^2$ + 8x + 3. $$What is the maximum height reached by the projectile?

Accepted Answer

$$y_{max} = 11$$

Question 8

Which of the following is the correct reciprocal identity for $$	an x$$?

Accepted Answer

$$	an x = \frac{1}{\cot x}$$

Question 9

A circle has the equation $$(x-2)^2$$ + $$(y+3)^2 = 25$. What are the center and radius of the circle?

Accepted Answer

Center $$(2, -3)$$, radius $$5$$

Question 10

Which of the following is the correct expression for $$\cos(x-y)$$?

Accepted Answer

$$\cos(x-y) = \cos x \cos y + \sin x \sin y$$

Question 11

A system of equations is given by $$\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}. $$What is the solution $$(x, y)$$?

Accepted Answer

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

Question 12

Which of the following is the correct Pythagorean identity?

Accepted Answer

$$\sin^2$ x + \cos^2$ x = 1$$

Question 13

A car's value depreciates according to $$V(t) = 20000e$$^{-0.15t}$$, where t$ is years. What is the value after 10 years?

Accepted Answer

$$V(10) = 20000e$$^{-1.5}$$ \approx 4462$$

Question 14

Which of the following is the correct formula for the equation of a hyperbola centered at the origin with transverse axis along the $$x-$$axis?

Accepted Answer

$$\frac{x^2$}{a^2$} - \frac{y^2$}{b^2$} = 1$$

Question 15

Which of the following is the correct expression for the inverse sine function?

Accepted Answer

$$y = \sin$$^{-1}$$ x$$ means $$\sin y = x$$ and $$-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$$

Question 16

A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?

Accepted Answer

$$\sqrt{6^2$ + 8^2$} = 10$$

Question 17

Which of the following is the correct double angle formula for cosine?

Accepted Answer

$$\cos(2x) = \cos^2$ x - \sin^2$ x$$

Question 18

A business's profit is modeled by $$P(x) = -3x^2$ + 12x - 7$$, where $$x is $$the number of units sold. How many units maximize the profit?

Accepted Answer

$$x = 2$$

Question 19

Which of the following is the correct expression for the reciprocal identity of $$\sec x$$?

Accepted Answer

$$\sec x = \frac{1}{\cos x}$$

Precalculus Study Notes: Functions, Trigonometry, and Analytic Geometry

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