Question 1

A company produces and sells a wireless doorbell with revenue given by $$R(x) = 81.5x$$ and cost given by $$C(x) = 58.5x + 2829$$, where $$x is $$the number of units produced and sold. What is the break-even number of units?

Accepted Answer

$$x = 131$$

Question 2

For the same company, what is the y-coordinate of the break-even point, and what does it represent?

Accepted Answer

$$y = 10,676.5$$, which represents the total revenue and total cost at break-even.

Question 3

Given the cost and revenue functions $$C(x) = 58.5x + 2829$$ and $$R(x) = 81.5x$$, what is the profit function $$P(x)$$?

Accepted Answer

$$P(x) = 23x - 2829$$

Question 4

A company’s profit from selling 232 units is $$321, and from 264 units is 545. $$What is the equation of the profit function $$P(x)$$?

Accepted Answer

$$P(x) = 7x - 1,293$$

Question 5

Using the profit function $$P(x) = 7x - 1,293$$, how many units must be produced and sold to avoid a loss?

Accepted Answer

$$x = 185$$

Question 6

Given the inequality $$-\frac{2}{3}(x-3) \u003e \frac{x}{3} + 1$$, what is the solution for $$x$$?

Accepted Answer

$$x \u003c 0$$

Question 7

Solve the system of equations by elimination: $$3x - 6y = -9$$ and $$9x = 15 - 3y. $$What is the solution $$(x, y)$$?

Accepted Answer

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

Question 8

What is the equation of the vertical line passing through the point $$(5, -7)$$?

Accepted Answer

$$x = 5$$

Question 9

What is the equation of the horizontal line passing through the point $$(9, 15)$$?

Accepted Answer

$$y = 15$$

Question 10

Given the following data for percent of U.S. residents who reported smoking: 27.0% in 1998, 25.8% in 2000, 25.0% in 2002, 24.9% in 2004, 25.1% in 2006, 25.4% in 2008, and 24.9% in 2010, what is the best linear model for $$y$$, the percent smoking, as a function of $$x$$, the number of years after 1985? (Round slope and intercept to three decimal places.)

Accepted Answer

$$y = -0.143x + 29.0$$

Question 11

In the context of the smoking model $$y = -0.143x + 29.0$$, what does the slope represent?

Accepted Answer

The percent of U.S. residents who smoke decreases by about 0.143% per year after 1985.

Question 12

Using the model $$y = -0.143x + 29.0$$, estimate the percent of U.S. residents who will smoke in 2025.

Accepted Answer

Approximately 23.4%

Question 13

Suppose a company’s cost function is $$C(x) = 58.5x + 2829$$ and its revenue function is $$R(x) = 81.5x. If $$the company sells 100 units, what is the profit?

Accepted Answer

$$P = 23x - 2829 = -529$$

Question 14

A company’s profit function is $$P(x) = 7x - 1,293. If $$the company wants a profit of at least $700, how many units must it sell?

Accepted Answer

$$x = 285$$

Question 15

If the cost to produce $$x$$ units is $$C(x) = 58.5x + 2829$$ and the revenue is $$R(x) = 81.5x$$, what is the fixed cost for the company?

Accepted Answer

$$2,829$$

Question 16

Given the system $$3x - 6y = -9$$ and $$9x = 15 - 3y$$, which of the following is the first step in the elimination method?

Accepted Answer

Rewrite the second equation as $$9x + 3y = 15$$

Question 17

A company’s revenue function is $$R(x) = 81.5x. $$What does the coefficient 81.5 represent in this context?

Accepted Answer

The price per unit sold

Question 18

If the cost function is $$C(x) = 58.5x + 2829$$, what is the variable cost per unit?

Accepted Answer

$$58.5$$

Question 19

Suppose the profit function is $$P(x) = 7x - 1,293. If $$the company sells 0 units, what is the profit and what does it represent?

Accepted Answer

$$-1,293$$, representing the fixed costs or initial loss

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