Q: A pharmaceutical company models the concentration of a drug in the bloodstream over time with the function $$C(t) = 50e$$^{-0.2t}$$, where t$ is in hours. What is the domain and range of C(t$$)$$?

Domain: $$[0, \\infty)$$; Range: $$(0, 50]$$

Q: A population of bacteria doubles every 3 hours. If the initial population is 500, which equation models the population $$P(t)$$ after $$t$$ hours?

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

Q: A medical imaging device uses the function $$y = 3 \\sin(2x - \\frac{\\pi}{4}) + 1 to $$model wave intensity. What is the amplitude, period, and phase shift of this function?

Amplitude: $$3$$, Period: $$\\pi$$, Phase shift: $$\\frac{\\pi}{8}$$ right

Q: A triangle in a bridge design has sides $$a = 8 m$$, $$b = 15 m$$, and included angle $$C = 60^\circ. $$What is the area of the triangle?

Area $$= \\frac{1}{2} \\cdot 8 \\cdot 15 \\cdot \\sin 60^\circ = 60 \\cdot \\frac{\\sqrt{3}}{2} = 30\\sqrt{3} m$^2$$

Q: A vector $$\\vec{v}$$ has components $$\\langle 3, 4 \\rangle. $$What is the magnitude of $$\\vec{v}$$, and what is the unit vector in the direction of $$\\vec{v}$$?

Magnitude: $$5$$, Unit vector: $$\\langle \\frac{3}{5}, \\frac{4}{5} \\rangle$$

Q: A savings account earns interest compounded continuously at 5% per year. If you deposit $$\1000$, what will the balance be after 10 years?

$$A = 1000 \\cdot e$$^{0.05 \\cdot 10}$$ = 1000 \\cdot e$$^{0.5}$$

Q: A geometric sequence models the decay of a radioactive substance. If the initial mass is $$200 g $$and the mass halves every hour, what is the mass after $$4$$ hours?

$$200 \\cdot (\\frac{1}{2})^4$ = 12.5 g$$

Q: A hyperbola is used to model the path of a comet. Its equation is $$\\frac{(x-1)^2$}{9} - \\frac{(y+2)^2$}{16} = 1. $$What are the coordinates of its center and the equations of its asymptotes?

Center: $$(1, -2)$$; Asymptotes: $$y + 2 = \\pm \\frac{4}{3}(x - 1)$$

Question 1

A pharmaceutical company models the concentration of a drug in the bloodstream over time with the function $$C(t) = 50e$$^{-0.2t}$$, where t$ is in hours. What is the domain and range of C(t$$)$$?

Accepted Answer

Domain: $$[0, \infty)$$; Range: $$(0, 50]$$

Question 2

A population of bacteria doubles every 3 hours. If the initial population is 500, which equation models the population $$P(t)$$ after $$t$$ hours?

Accepted Answer

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

Question 3

A quadratic function $$f(x) = ax^2$ + bx + c$$ has a discriminant $$b^2 - 4ac = 0$. What does this tell you about the roots of the equation f(x) = 0$?

Accepted Answer

There is one real root (a repeated root).

Question 4

A rational function $$f(x) = \frac{(x-2)(x+1)}{(x-2)(x-3)} is $$used to model a chemical reaction rate. What feature occurs at $$x = 2$$?

Accepted Answer

A hole at $$x = 2$$

Question 5

A medical imaging device uses the function $$y = 3 \sin(2x - \frac{\pi}{4}) + 1 to $$model wave intensity. What is the amplitude, period, and phase shift of this function?

Accepted Answer

Amplitude: $$3$$, Period: $$\pi$$, Phase shift: $$\frac{\pi}{8}$$ right

Question 6

A triangle in a bridge design has sides $$a = 8 m$$, $$b = 15 m$$, and included angle $$C = 60^\circ. $$What is the area of the triangle?

Accepted Answer

Area $$= \frac{1}{2} \cdot 8 \cdot 15 \cdot \sin 60^\circ = 60 \cdot \frac{\sqrt{3}}{2} = 30\sqrt{3} m$^2$$

Question 7

A vector $$\vec{v}$$ has components $$\langle 3, 4 \rangle. $$What is the magnitude of $$\vec{v}$$, and what is the unit vector in the direction of $$\vec{v}$$?

Accepted Answer

Magnitude: $$5$$, Unit vector: $$\langle \frac{3}{5}, \frac{4}{5} \rangle$$

Question 8

A circle is used to model the cross-section of a pipe. Its equation is $$(x - 2)^2$ + (y + 3)^2$ = 16. $$What is the center and radius of the pipe?

Accepted Answer

Center: $$(2, -3)$$; Radius: $$4$$

Question 9

A savings account earns interest compounded continuously at 5% per year. If you deposit $$\1000$, what will the balance be after 10 years?

Accepted Answer

$$A = 1000 \cdot e$$^{0.05 \cdot 10}$$ = 1000 \cdot e$$^{0.5}$$

Question 10

A geometric sequence models the decay of a radioactive substance. If the initial mass is $$200 g $$and the mass halves every hour, what is the mass after $$4$$ hours?

Accepted Answer

$$200 \cdot (\frac{1}{2})^4$ = 12.5 g$$

Question 11

A system of equations models the intersection of two roads: $$y = 2x + 3$$ and $$y = -x + 1. $$What is the point of intersection?

Accepted Answer

$$(x, y) = (2/3, 7/3)$$

Question 12

A hyperbola is used to model the path of a comet. Its equation is $$\frac{(x-1)^2$}{9} - \frac{(y+2)^2$}{16} = 1. $$What are the coordinates of its center and the equations of its asymptotes?

Accepted Answer

Center: $$(1, -2)$$; Asymptotes: $$y + 2 = \pm \frac{4}{3}(x - 1)$$

Question 13

A trigonometric equation $$2 \sin^2$ x - \sin x - 1 = 0$$ arises in modeling a pendulum. What are the solutions for $$x in$$ $$[0, 2\pi)$$?

Accepted Answer

$$x = \frac{\pi}{2}, \frac{7\pi}{6}, \frac{11\pi}{6}$$

Question 14

A piecewise function models the temperature in a climate-controlled room: $$f(x) = \begin{cases} 20, & x \u003c 8 \ 25, & 8 \leq x \u003c 18 \ 22, & x \geq 18 \end{cases}$$, where $$x is $$the hour of the day. What is $$f(9)$$ and $$f(19)$$?

Accepted Answer

$$f(9) = 25$$, $$f(19) = 22$$

Question 15

A vector $$\vec{u} = \langle 2, -1 \rangle$$ and $$\vec{v} = \langle -4, 2 \rangle$$ represent forces acting on a structure. Are these vectors parallel, orthogonal, or neither?

Accepted Answer

Orthogonal, because $$\vec{u} \cdot \vec{v} = 0$$

Question 16

A sequence models the number of seats in each row of a theater: Row 1 has 20 seats, each subsequent row has 3 more seats than the previous. What is the number of seats in Row 10?

Accepted Answer

$$a_{10} = 20 + (10 - 1) \cdot 3 = 47$$

Question 17

A logarithmic equation $$\log_2$(x - 1) = 3$$ models the pH of a solution. What is the value of $$x$$?

Accepted Answer

$$x = 9$$

Question 18

A parabola models the trajectory of a thrown ball: $$(x - 2)^2$ = 12(y + 1). $$What is the vertex and focus of the parabola?

Accepted Answer

Vertex: $$(2, -1)$$; Focus: $$(2, 2)$$

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