Chapter 11, Problem 29CCSR

To determine

Find the total cost of labor for a repair in $\text{4}\frac{1}{2}$ hours.

Answer to Problem 29CCSR

Total cost of labor for a repair in $\text{4}\frac{1}{2}$ hours is $435.

Explanation of Solution

Given:

The graph below shows the piecewise function for the total cost for x hours of labor at an automotive repair center. What will be the total cost of labor for a repair that takes $\text{4}\frac{1}{2}$ hours?

The graph:

  

Concept Used:

Mathematically speaking, a step function is a function whose graph looks like a series of steps because it consists of a series of horizontal line segments with jumps in-between. For this reason, it is also sometimes called a staircase function.

A step function has constant value on given intervals, but the constant is different for each interval. The constant value on each interval creates the series of horizontal lines, and the fact that the constant is different for each interval creates the jumps in between each horizontal line segment. This is why the graph of a step function looks like a set of stairs.

For example, the piecewise function given by

  

This function is defined by two equations.

One equation gives the values of $f(x)$ ) when $x\le 1$ , and the other equation gives the values of $f(x)$ when $x>1$ .

Calculation:

Total cost of 1 hour = $50 Total cost of 2 hour = $85 Total cost of 3 hour = $120 Total cost of $\text{4}\frac{1}{2}$ hour = $180 Total cost of labor in $\text{4}\frac{1}{2}$ hours is: $\$\left(50+85+120+180\right)=\$435$

Thus, total cost of labor for a repair in $\text{4}\frac{1}{2}$ hours is $435.