Q: A company’s demand function is given by $$p = 50 - 2x$$, where $$p is $$the price per unit and $$x is $$the number of units sold. What is the total revenue $$R$$ when 10 units are sold?

$$R = (50 - 2 \times 10) \times 10 = 300$$

Q: Given the cost function $$C(x) = 5x + 200$$, what is the average cost per unit when 20 units are produced?

$$C = \frac{5 \times 20 + 200}{20} = 15$$

Q: Which of the following is the correct application of the quadratic formula to solve $$2x^2 - 4x - 6 = 0$?

$$x = \frac{4 \pm \sqrt{(-4)^2$ - 4 \times 2 \times (-6)}}{2 \times 2}$$

Q: Which of the following is the correct way to rationalize the denominator of $$\frac{5}{\sqrt{3}}$$?

$$\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$$

Q: If $$f(x) = x^2$ - 4x + 7$$, what is $$f(3)$$?

$$f(3) = 3^2$ - 4 \times 3 + 7 = 2$$

Question 1

A company’s demand function is given by $$p = 50 - 2x$$, where $$p is $$the price per unit and $$x is $$the number of units sold. What is the total revenue $$R$$ when 10 units are sold?

Accepted Answer

$$R = (50 - 2 	imes 10) 	imes 10 = 300$$

Question 2

Which of the following is the correct factorization of $$x^2 - 16$?

Accepted Answer

$$(x - 4)(x + 4)$$

Question 3

Given the cost function $$C(x) = 5x + 200$$, what is the average cost per unit when 20 units are produced?

Accepted Answer

$$C = \frac{5 	imes 20 + 200}{20} = 15$$

Question 4

Simplify the expression $$\frac{2x^2$y}{4xy^2$} to $$lowest terms.

Accepted Answer

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

Question 5

If $$x^2 = 25$, what are the possible values of x$?

Accepted Answer

$$x = \pm 5$$

Question 6

Which of the following is equivalent to $$\frac{a}{b} \div \frac{c}{d}$$?

Accepted Answer

$$\frac{ad}{bc}$$

Question 7

Find the slope of the line passing through the points $$(2, 5)$$ and $$(6, 13).$$

Accepted Answer

$$m = \frac{13 - 5}{6 - 2} = 2$$

Question 8

Which of the following is the correct application of the quadratic formula to solve $$2x^2 - 4x - 6 = 0$?

Accepted Answer

$$x = \frac{4 \pm \sqrt{(-4)^2$ - 4 	imes 2 	imes (-6)}}{2 	imes 2}$$

Question 9

A business has a revenue function $$R(x) = 20x$$ and a cost function $$C(x) = 8x + 120. $$What is the profit function $$P(x)$$?

Accepted Answer

$$P(x) = 20x - (8x + 120) = 12x - 120$$

Question 10

Which of the following is the correct way to rationalize the denominator of $$\frac{5}{\sqrt{3}}$$?

Accepted Answer

$$\frac{5}{\sqrt{3}} 	imes \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$$

Question 11

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

Accepted Answer

$$x 
eq 2$$

Question 12

Which of the following is the correct factorization of $$x^2 + 6x + 9$?

Accepted Answer

$$(x + 3)^2$

Question 13

A company’s supply and demand functions are $$S(x) = 2x + 10$$ and $$D(x) = 50 - x. $$Find the equilibrium quantity $$x_0$$ where supply equals demand.

Accepted Answer

$$2x + 10 = 50 - x \implies 3x = 40 \implies x_0$ = \frac{40}{3}$$

Question 14

Simplify the expression $$\frac{3x^2$y^3$$}{9xy}$.

Accepted Answer

$$\frac{y^2$x}{3}$$

Question 15

If $$f(x) = x^2$ - 4x + 7$$, what is $$f(3)$$?

Accepted Answer

$$f(3) = 3^2$ - 4 	imes 3 + 7 = 2$$

Question 16

Which of the following is the correct simplification of $$\frac{2}{\sqrt{5} - 1}$$?

Accepted Answer

$$\frac{2(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)} = \frac{2(\sqrt{5} + 1)}{5 - 1} = \frac{2(\sqrt{5} + 1)}{4}$$

Question 17

Which of the following is NOT a valid property of exponents?

Accepted Answer

$$x^n + x^m = x$$^{n+m}$$

Question 18

A company’s cost function is $$C(x) = 4x + 100$$ and its revenue function is $$R(x) = 10x. $$What is the break-even quantity?

Accepted Answer

$$4x + 100 = 10x \implies 6x = 100 \implies x = \frac{100}{6}$$

Question 19

Which of the following is the correct point-slope form for the line passing through $$(1, 2)$$ with slope $$m = -3$$?

Accepted Answer

$$y - 2 = -3(x - 1)$$

College Algebra: Essential Formulas and Concepts for Business Applications

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