Q. 53 - Precalculus Enhanced with Graphing Utilities

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.

Step-by-Step Solution

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 $C\left(x\right)=170+0.25(x-400)$

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 $100 and 300$ miles and the remaining miles.

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