Chapter 2.4, Problem 53AYU

Cost of Transporting Goods 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, $\$0.50$ per mile for the first 100 miles, $\$0.40$ 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.

To determine

To find:

a. Graph the relationship between the cost of transportation in dollars and mileage over the entire 960-mile route.

Answer to Problem 53AYU

Solution:

a.

Explanation of Solution

Given:

It is given that the trucking company transports goods between Chicago and New York of 960 miles.

The company charges $\$0.50$ per mile for the first 100 miles.

It charges $\$0.40$ per mile for the next 300 miles (100 to 400 miles).

Then it charges $\$0.25$ per mile for the next 400 miles (400 to 800 miles) and

There is no charge for the remaining 160 miles.

Therefore there will be four limiting conditions for the transport cost $x$. They are:

1. $x \le 100$

2. $100 < x \le 400$

3. $400 < x \le 800$

4. $800 < x \le 960$

For the condition $x \le 100$,

As the cost is $\$0.50$ per mile for the first 100 miles, the function becomes

$0.5x$   if   $x \le 100$

For the condition $100 < x \le 400$,

As the cost is $\$0.40$ per mile for next 300 miles, the function becomes

$0.5\left(100\right) + \left(x - 100\right)0.4$   if   $100 < x \le 400$

$10 + 0.4x$   if   $100 < x \le 400$

For the condition $400 < x \le 800$,

As the cost is $\$0.25$ per mile for next 400 miles, the function becomes

$0.5\left(100\right) + 0.4\left(300\right) + \left(x - 400\right)0.25$   if   $400 < x \le 800$

$70 + 0.25x$   if$400 < x \le 800$

For the condition $800 < x \le 960$,

As there is no cost for next 160 miles, the function becomes

$0.5\left(100\right) + 0.4\left(300\right) + 0.25\left(400\right) + 0\left(160\right)$   if   $800 < x \le 960$

$270$   if   $800 < x \le 960$

Let the function for the cost be $C\left(x\right)$. The function can be defined as

$C\left(x\right) = \{\begin{array}{l}0.5xif x\le 100\\ 10+0.4x if 100<x\le 400\\ 70+0.25x if 400<x\le 800\\ 270 if 800<x\le 960\end{array}$

a.

To determine

To find:

b. The cost as a function of mileage for hauls between 100 and 400 miles from Chicago.

Answer to Problem 53AYU

Solution:

b. The cost as a function of mileage for hauls between 100 and 400 miles from Chicago is $10 + 0.4x$.

Explanation of Solution

Given:

It is given that the trucking company transports goods between Chicago and New York of 960 miles.

The company charges $\$0.50$ per mile for the first 100 miles.

It charges $\$0.40$ per mile for the next 300 miles (100 to 400 miles).

Then it charges $\$0.25$ per mile for the next 400 miles (400 to 800 miles) and

There is no charge for the remaining 160 miles.

Therefore there will be four limiting conditions for the transport cost $x$. They are:

5. $x \le 100$

6. $100 < x \le 400$

7. $400 < x \le 800$

8. $800 < x \le 960$

For the condition $x \le 100$,

As the cost is $\$0.50$ per mile for the first 100 miles, the function becomes

$0.5x$   if   $x \le 100$

For the condition $100 < x \le 400$,

As the cost is $\$0.40$ per mile for next 300 miles, the function becomes

$0.5\left(100\right) + \left(x - 100\right)0.4$   if   $100 < x \le 400$

$10 + 0.4x$   if   $100 < x \le 400$

For the condition $400 < x \le 800$,

As the cost is $\$0.25$ per mile for next 400 miles, the function becomes

$0.5\left(100\right) + 0.4\left(300\right) + \left(x - 400\right)0.25$   if   $400 < x \le 800$

$70 + 0.25x$   if   $400<x\le 800$

For the condition $800 < x \le 960$,

As there is no cost for next 160 miles, the function becomes

$0.5\left(100\right) + 0.4\left(300\right) + 0.25\left(400\right) + 0\left(160\right)$   if   $800 < x \le 960$

$270$   if   $800 < x \le 960$

Let the function for the cost be $C\left(x\right)$. The function can be defined as

$C\left(x\right) = \{\begin{array}{l}0.5xif x\le 100\\ 10+0.4x if 100<x\le 400\\ 70+0.25x if 400<x\le 800\\ 270 if 800<x\le 960\end{array}$

b. From the derived function, it can be conclude that the cost as a function of mileage for hauls between 100 and 400 miles from Chicago is $10 + 0.4x$.

To determine

To find:

c. The cost as a function of mileage for hauls between 400 and 800 miles from Chicago.

Answer to Problem 53AYU

Solution:

c. The cost as a function of mileage for hauls between 400 and 800 miles from Chicago is $\underset{\_}{70 + 0.25x}$

Explanation of Solution

Given:

It is given that the trucking company transports goods between Chicago and New York of 960 miles.

The company charges $\$0.50$ per mile for the first 100 miles.

It charges $\$0.40$ per mile for the next 300 miles (100 to 400 miles).

Then it charges $\$0.25$ per mile for the next 400 miles (400 to 800 miles) and

There is no charge for the remaining 160 miles.

Therefore there will be four limiting conditions for the transport cost $x$. They are:

9. $x \le 100$

10. $100 < x \le 400$

11. $400 < x \le 800$

12. $800 < x \le 960$

For the condition $x \le 100$,

As the cost is $\$0.50$ per mile for the first 100 miles, the function becomes

$0.5x$   if   $x \le 100$

For the condition $100 < x \le 400$,

As the cost is $\$0.40$ per mile for next 300 miles, the function becomes

$0.5\left(100\right) + \left(x - 100\right)0.4$   if   $100 < x \le 400$

$10 + 0.4x$   if   $100 < x \le 400$

For the condition $400 < x \le 800$,

As the cost is $\$0.25$ per mile for next 400 miles, the function becomes

$0.5\left(100\right) + 0.4\left(300\right) + \left(x - 400\right)0.25$   if   $400 < x \le 800$

$70 + 0.25x$   if$400 < x \le 800$

For the condition $800 < x \le 960$,

As there is no cost for next 160 miles, the function becomes

$0.5\left(100\right) + 0.4\left(300\right) + 0.25\left(400\right) + 0\left(160\right)$   if   $800 < x \le 960$

$270$   if   $800 < x \le 960$

Let the function for the cost be $C\left(x\right)$. The function can be defined as

$C\left(x\right) = \{\begin{array}{l}0.5xif x\le 100\\ 10+0.4x if 100<x\le 400\\ 70+0.25x if 400<x\le 800\\ 270 if 800<x\le 960\end{array}$

c. From the derived function, it can be conclude that the cost as a function of mileage for hauls between 400 and 800 miles from Chicago is $70 + 0.25x$.