Chapter 4.7, Problem 42PPS

To determine

To Graph:

Draw a graph to represent the cost of going over the mileage.

Explanation of Solution

Given Information:

As per Marcus’ leasing agreement , he will be charged $0.20 per mile for each mile over 12,000.Any fraction of a mile is rounded up to the next mile.

Concept Used:

Least Integer Function:

The function whose value at any number x is the smallest integer greater than of equal to x is called the least integer function.

It is denoted by $\lceil x\rceil$ . It is also known as ceiling of x .

For example

  $\begin{array}{l}\lceil 3.578\rceil =4\\ \lceil 0.78\rceil =1\\ \lceil -4.64\rceil =-4\end{array}$

Calculation:

Charges per mile above 12,000 miles = $0.20

For any fraction of a mile above 12,000 miles , we round it up to the next mile and then take the following steps ( for example , for 0.6 miles , we take number of miles , x = 1) :

So , for x miles above 12,000 miles , we take ceiling of x = $\lceil x\rceil$ .

So, for x miles above 12,000 miles, we have the charges will be applied = $\lceil x\rceil \cdot (0.20)$

Cost of going x miles above 12,000 miles = $f(x)=\lceil x\rceil \cdot (0.20)$

We make a table for different number of miles above 12,000 miles and Cost of going above the mileage:

Number of miles over 12,000 (x)	Rounded up value = $\lceil x\rceil$	$f(x)=\lceil x\rceil \cdot (0.20)$
1	1	0.20
1.5	2	0.40
2	2	0.40
2.5	3	0.60
3	3	0.60
3.5	4	0.80
4	4	0.80

Graph:

From the table above , we draw the graph of the function :

  $f(x)=\lceil x\rceil \cdot (0.20)$

Where to use the dots and circles :

At 12,000 miles, the cost is included in rental, so it is a circle at (0,0.20).

From 12,000 miles (excluded ) to 12,001 miles (included) , there is a cost of $0.20.

So, at x = 1( that is at 12,001 miles) , the cost is $0.20.

So, at (0,0.20) , there is a circle and at (1,0.20) , there is dot.

From 12,001 ( x =1) to 12,002 ( x = 2 ) miles , the cost is $0.40 , but 12,001 not included and 12,002 is included.

So, at (1,0.40) , there is a circle and at (2,0.40) , there is dot.

Plot the points and join.Continue the proess.