Question

A trucking company transports goods between Chicago and New York, a distance of $960$miles. The company’s policy is to charge, for each pound, $\backslash (0.50$per mile for the first 100 miles, $\backslash )0.40$per mile for the next $300$ miles, $\$0.25$per mile for the next $400$ miles,
and no charge for the remaining $160$miles.

(a) Graph the relationship between the cost of transportation in dollars and mileage over the entire $960$ mile route.
(b) Find the cost as a function of mileage for hauls between $100$and $400$ miles from Chicago.
(c) Find the cost as a function of mileage for hauls between
$400$ and $800$ miles from Chicago.

Short answer

(a) The relationship between the cost of transportation in dollars and mileage over the entire route is:

(b) The function of mileage for hauls between $100$and $400$miles from Chicago is $C\left(x\right)=50=0.4(x-100)$

(c) The function of mileage for hauls between $400$and $800$ miles from Chicago is $C\left(x\right)=170+0.25(x-400)$

Step-by-step solution

Part (a) Step 1. Given

$400$The distance between Chicago and New York is $960$miles.

The charge is $\$0.50$for the first $100$miles

The charge is $\$0.40$for the next $300$miles

The charge is $\$0.25$for the next $400$miles

No charge for last $160$miles.

To graph the relationship between the cost of transportation in dollars and mileage over the entire route.

Part (a) Step 2. Find the cost for transportation

For $100$miles, the charge is $\$0.50$per mile, so the charge is $\$50$

For the next $300$miles, the sum of charge for the first $100$along with the charge of next $300$miles, so $50+0.40\left(300\right)=\$170$

Similarly the charge for next $400$miles is$=50+120+0.25\left(400\right)$

$=\$270$

Similarly the charge for next $160$miles remains unchanged.

So, the charge for next $160$ miles is $\$270$

Part (a) Step 3. Sketch the graph.

Plot the points $(100,50),(400,170),(800,270),(960,270)$in the graph.

Part (b) Step 1. To find

To find the cost as a function of mileage for hauls between $100$and $400$miles from Chicago.

Part (b) Step 2. Find the function

The cost of hauls between $100$and $400$miles is the sum of the cost for first $100$miles and that of the remaining hauls.

If $x$represent the haul between $100$and $400$, then,

$C=50+0.4(x-100)$

Part (c) Step 1. To find

To find the cost function of mileage for hauls between $400$and $800$ miles from Chicago.

Part (c) Step 2. Find he function

The cost for the hauls between $400$and $800$hauls is the sum of the cost for the first $100and300$miles and the remaining miles.

$$\begin{gathered} C\left(x\right)=50+120+0.25(x-400) \\ =170+0.25(x-400) \end{gathered}$$