Question

1. maria knows that every day her electrical shop is open, the overhead costs are $167.52. (overhead costs include heating, air-conditioning, lighting, insurance, etc.) in addition, she has to pay $74.32 per hour in wages to her employees. you want to write an equation for her costs, \\(c\\), each hour, \\(h\\), of an eight-hour day.

(a) what is the \\(c\\)-intercept?
(b) what is the slope?
(c) what is the equation?
(d) how much must she make in a typical day in order to not lose money?

1. challenge: a line has an \\(x\\)-intercept of \\(-4\\) and a \\(y\\)-intercept of \\(-3\\). write an equation of the line.

Explanation:

Identify the parameters for Question 20

Using the Linear Modeling knowledge point

• Daily fixed overhead cost: \(167.52\) dollars
• Hourly variable wage cost: \(74.32\) dollars per hour
• Independent variable: \(h\) (hours, where \(0 \le h \le 8\))
• Dependent variable: \(C\) (total cost in dollars)

Determine the intercept and slope for Question 20

Using the y-intercept and Slope Formula knowledge points

• The \(C\)-intercept represents the cost when \(h = 0\), which is the fixed daily overhead:

$$ C\text{-intercept} = 167.52 $$

• The slope represents the rate of change of cost per hour, which is the hourly wage rate:

$$ \text{Slope} = 74.32 $$

Formulate the cost equation and calculate the daily break-even cost

Using the Equation of a Line knowledge point

• The linear equation relating cost \(C\) and hours \(h\) is:

$$ C = 74.32h + 167.52 $$

• To find how much she must make in an eight-hour day (\(h = 8\)) to not lose money, we calculate the total cost \(C\) at \(h = 8\):

$$ C = 74.32(8) + 167.52 = 594.56 + 167.52 = 762.08 $$

Solve the Challenge Question 21

Using the y-intercept and Slope Formula knowledge points

• Given intercepts: \((-4, 0)\) and \((0, -3)\)
• Calculate the slope \(m\):

$$ m = \frac{-3 - 0}{0 - (-4)} = -\frac{3}{4} $$

• Write the equation in slope-intercept form using the \(y\)-intercept \(b = -3\):

$$ y = -\frac{3}{4}x - 3 $$

Answer:

Question 20

(a) The \(C\)-intercept is \(167.52\) (or the point \((0, 167.52)\)).

(b) The slope is \(74.32\).

(c) The equation is \(C = 74.32h + 167.52\).

(d) She must make \(\$762.08\) in a typical eight-hour day to not lose money.

Question 21

The equation of the line is \(y = -\frac{3}{4}x - 3\) (or in standard form, \(3x + 4y = -12\)).