Q: A pharmaceutical company models the decay of a drug in the bloodstream using the formula $$A = A_0$ e$$^{-kt}$$, where A_0$ is the initial amount, k$ is the decay constant, and t$ is time in hours. If A_0$ = 100 mg$$, $$k = 0.2$$, and $$t = 5$$ hours, what is the amount of drug remaining?

$$A = 100 e$$^{-0.2 \times 5}$$ = 100 e$$^{-1}$$ \approx 36.79 mg$$

Q: A quadratic equation models the height $$h (in $$meters) of a ball thrown upward: $$h(t) = -4.9t^2$ + 20t + 2. $$What is the maximum height reached by the ball?

The vertex occurs at $$t = \frac{-b}{2a} = \frac{-20}{2 \times -4.9} \approx 2.04 s$$; $$h(2.04) \approx 22.4 m$$

Q: A hospital records the number of patients admitted each day as a sequence: 12, 15, 18, 21. What is the formula for the $$nth $$term and what is the 10th term?

$$a_n = 12 + 3(n-1);\ a_{10} = 12 + 3 \times 9 = 39$$

Q: A company’s profit is modeled by $$P(x) = 5x^2$ - 20x + 15. $$Using the quadratic formula, what are the values of $$x$$ where profit is zero?

$$x = \frac{20 \pm \sqrt{(-20)^2$ - 4 \times 5 \times 15}}{2 \times 5} = \frac{20 \pm \sqrt{400 - 300}}{10} = \frac{20 \pm 10}{10}$$, so $$x = 3$$ and $$x = 1$$

Q: A biologist models population growth with $$P(t) = 500 e$$^{0.05t}$$. What is the population after 10 years?

$$P(10) = 500 e$$^{0.5}$$ \approx 500 \times 1.6487 \approx 824$$

Q: A company’s revenue is modeled by $$R(x) = 3x^2$ - 12x + 9. $$What is the end behavior of $$R(x) as$$ $$x \to \infty$$?

As $$x \to \infty$$, $$R(x) \to \infty ($$since leading coefficient is positive)

Q: A geometric sequence models the spread of a rumor: 5, 10, 20, 40. What is the common ratio and the formula for the $$nth $$term?

Common ratio $$r = 2$$, $$a_n = 5 \times 2$$^{n-1}$$

Question 1

A pharmaceutical company models the decay of a drug in the bloodstream using the formula $$A = A_0$ e$$^{-kt}$$, where A_0$ is the initial amount, k$ is the decay constant, and t$ is time in hours. If A_0$ = 100 mg$$, $$k = 0.2$$, and $$t = 5$$ hours, what is the amount of drug remaining?

Accepted Answer

$$A = 100 e$$^{-0.2 	imes 5}$$ = 100 e$$^{-1}$$ \approx 36.79 mg$$

Question 2

A quadratic equation models the height $$h (in $$meters) of a ball thrown upward: $$h(t) = -4.9t^2$ + 20t + 2. $$What is the maximum height reached by the ball?

Accepted Answer

The vertex occurs at $$t = \frac{-b}{2a} = \frac{-20}{2 	imes -4.9} \approx 2.04 s$$; $$h(2.04) \approx 22.4 m$$

Question 3

Given the function $$f(x) = \frac{x-1}{x^2$-4}$$, which of the following describes its vertical asymptotes?

Accepted Answer

Vertical asymptotes at $$x = 2$$ and $$x = -2$$

Question 4

A hospital records the number of patients admitted each day as a sequence: 12, 15, 18, 21. What is the formula for the $$nth $$term and what is the 10th term?

Accepted Answer

$$a_n = 12 + 3(n-1);\ a_{10} = 12 + 3 	imes 9 = 39$$

Question 5

A company’s profit is modeled by $$P(x) = 5x^2$ - 20x + 15. $$Using the quadratic formula, what are the values of $$x$$ where profit is zero?

Accepted Answer

$$x = \frac{20 \pm \sqrt{(-20)^2$ - 4 	imes 5 	imes 15}}{2 	imes 5} = \frac{20 \pm \sqrt{400 - 300}}{10} = \frac{20 \pm 10}{10}$$, so $$x = 3$$ and $$x = 1$$

Question 6

A biologist models population growth with $$P(t) = 500 e$$^{0.05t}$$. What is the population after 10 years?

Accepted Answer

$$P(10) = 500 e$$^{0.5}$$ \approx 500 	imes 1.6487 \approx 824$$

Question 7

A piecewise function models a parking fee: $$f(x) = \begin{cases} 2x & 	ext{if } x \leq 3 \ 6 + (x-3) & 	ext{if } x \u003e 3 \end{cases}$$, where $$x is $$hours parked. What is the fee for 5 hours?

Accepted Answer

$$f(5) = 6 + (5-3) = 8$$

Question 8

A chemical reaction’s rate is modeled by $$R(x) = x^3$ - 4x. $$What is the multiplicity of the zero at $$x = 0$$?

Accepted Answer

Multiplicity is 1 (since $$x^3 - 4x$$ = $$x(x^2 - 4$$)$$, x=0$ is a single root)

Question 9

A student solves the system: $$\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$$ using substitution. What is the solution?

Accepted Answer

Substitute $$x = y + 1$$ into first: $$2(y+1) + 3y = 7 \rightarrow 2y + 2 + 3y = 7 \rightarrow 5y = 5 \rightarrow y = 1$$, $$x = 2. $$Solution: $$(2, 1)$$

Question 10

A function $$f(x) = x^2$ + 10 is $$reflected over the $$x-$$axis and shifted down by 4 units. What is the new function?

Accepted Answer

$$f(x) = -x^2$ + 6$$

Question 11

A company’s revenue is modeled by $$R(x) = 3x^2$ - 12x + 9. $$What is the end behavior of $$R(x) as$$ $$x 	o \infty$$?

Accepted Answer

As $$x 	o \infty$$, $$R(x) 	o \infty ($$since leading coefficient is positive)

Question 12

A geometric sequence models the spread of a rumor: 5, 10, 20, 40. What is the common ratio and the formula for the $$nth $$term?

Accepted Answer

Common ratio $$r = 2$$, $$a_n = 5 	imes 2$$^{n-1}$$

Question 13

A matrix $$A = \begin{pmatrix} 2 & 3 \ 1 & 4 \end{pmatrix}$$ represents a transformation. What is its determinant?

Accepted Answer

$$\det(A) = 2 	imes 4 - 3 	imes 1 = 8 - 3 = 5$$

Question 14

A scientist uses the logarithmic rule to convert $$\log_{10} 1000 to $$exponential form. What is the equivalent expression?

Accepted Answer

$$\log_{10} 1000 = 3$$ because $$10^3 = 1000$

Question 15

A quadratic equation $$x^2 - 6x + 9 = 0$ is solved using the square root property. What are the solutions?

Accepted Answer

$$x^2 - 6x + 9$$ = $$(x-3)^2 = 0$$ \rightarrow x-3 = 0 \rightarrow x = 3$$

Question 16

A linear inequality $$5x \u003e 20$$ models a budget constraint. What is the solution set?

Accepted Answer

$$x \u003e 4$$

Question 17

A function $$f(x) = x^2$ is tested for symmetry. Which type of symmetry does it have?

Accepted Answer

It has $$y-$$axis symmetry (even function)

Question 18

A sequence is defined by $$a_n = 2 + 4(n-1). $$What is the sum of the first 4 terms?

Accepted Answer

First 4 terms: $$2, 6, 10, 14. $$Sum: $$2 + 6 + 10 + 14 = 32$$

Question 19

A rational function $$f(x) = \frac{3x^2$ - 2x + 1}{x^2$ + 1} is $$evaluated as $$x 	o \infty. $$What is the horizontal asymptote?

Accepted Answer

Horizontal asymptote at $$y = 3 ($$ratio of leading coefficients)

College Algebra Final Exam Study Guide

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