Question

a plumber charges a $50 service fee plus $75 per hour. the cost function is: c(h)=75h + 50. questions: 1. graph the function for 0 to 5 hours. 2. what is the y - intercept and what does it represent? 3. what is the slope and what does it represent? 4. after how many hours will the total cost be $275?

Explanation:

Step1: Identify the function form

The cost - function is $C(h)=75h + 50$, which is in the slope - intercept form $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.

Step2: Find the y - intercept and its meaning

For the function $C(h)=75h + 50$, when $h = 0$, $C(0)=50$. The y - intercept is 50, and it represents the fixed service fee of $50$ that the plumber charges regardless of the number of hours worked.

Step3: Find the slope and its meaning

The slope of the function $C(h)=75h + 50$ is 75. It represents the rate of change of the cost with respect to the number of hours. In other words, the plumber charges $75$ per hour of work.

Step4: Solve for the number of hours when the cost is $275$

Set $C(h)=275$ in the equation $C(h)=75h + 50$. So, $275=75h + 50$.
Subtract 50 from both sides: $275−50 = 75h$, which gives $225 = 75h$.
Divide both sides by 75: $h=\frac{225}{75}=3$.

Step5: Graph the function

To graph $C(h)=75h + 50$ for $0\leq h\leq5$:

• When $h = 0$, $C(0)=50$. So, the y - intercept is the point $(0,50)$.
• When $h = 1$, $C(1)=75\times1 + 50=125$, the point is $(1,125)$.
• When $h = 2$, $C(2)=75\times2+50 = 200$, the point is $(2,200)$.
• When $h = 3$, $C(3)=75\times3 + 50=275$, the point is $(3,275)$.
• When $h = 4$, $C(4)=75\times4+50 = 350$, the point is $(4,350)$.
• When $h = 5$, $C(5)=75\times5+50 = 425$, the point is $(5,425)$. Plot these points and draw a straight line through them for the domain $0\leq h\leq5$.

Answer:

1. To graph the function $C(h)=75h + 50$ for $0\leq h\leq5$, plot the points $(0,50),(1,125),(2,200),(3,275),(4,350),(5,425)$ and draw a straight - line through them.
2. The y - intercept is 50. It represents the fixed service fee of $50$ that the plumber charges regardless of the number of hours worked.
3. The slope is 75. It represents the rate of $75$ per hour that the plumber charges for work.
4. After 3 hours, the total cost will be $275$.