Chapter 2.4, Problem 50AYU

(a)

To determine

To find: the charge for using 40 therms in a month.

Answer to Problem 50AYU

The charge for using 40 therms in one month is $29.704.

Explanation of Solution

Given information:

Monthly customer charge $8.40.

Per therm service charge

1^st 20 therms $0.1473 /therm

Next 30 therms $0.0579/therm

Over 50 therms  $0.0519/therm

Gas supply charge  $0.43/therm

Calculation:

For using 40 therms in a month, the rate of distribution charge $0.1473 /therm applied for inItial 20 therms.

The rate of distribution charge $0.0579/therm applied for next 20 therms.

In addition monthly customer charge $8.40 and gas charge $0.43/therm also applied.

Gas supply charge

  $40\times \$0.43=\$17.20$

Distribution charge

  $20\times \$0.1473+20\times \$0.0579=\$4.104$

Total charge

  $\$8.40+\$17.20+\$4.104=\$29.704$

Therefore, the charge for using 40 therms in one month is $\$29.704$ .

(b)

To determine

To find: the charge for using 150 therms in a month.

Answer to Problem 50AYU

The charge for using 150 therms in one month is $\$129.483$ .

Explanation of Solution

Given information:

Monthly customer charge $8.40.

Per therm service charge

1^st 20 therms $0.1473 /therm

Next 30 therms $0.0579/therm

Over 50 therms  $0.0519/therm

Gas supply charge  $0.43/therm

Calculation:

For using 150 therms in a month, the rate of distribution charge $0.1473 /therm applied for inItial 20 therms.

The rate of distribution charge $0.0579/therm applied for next 30 therms.

The rate of distribution charge $0.0519/therm applied for next 100 therms.

In addition monthly customer charge $8.40 and gas charge $0.43/therm also applied.

Gas supply charge

  $150\times \$0.43=\$64.50$

Distribution charge

  $20\times \$0.1473+30\times \$0.0579+100\times \$0.0519=\$56.583$

Total charge

  $\$8.40+\$64.50+\$56.583=\$129.483$

Therefore, the charge for using 150 therms in one month is $\$129.483$ .

(c)

To determine

To develop: a model that relates the monthly charge $C$ for $x$ therms of gas.

Answer to Problem 50AYU

  $C\left(x\right)=\{\begin{array}{l}\$0.5773x+\$8.40if0\le x\le 20\\ \$0.4879x+\$10.188if20\le x<50\\ \$0.4819x+\$10.488ifx>50\end{array}$

Explanation of Solution

Given information:

Monthly customer charge $8.40.

Per therm service charge

1^st 20 therms $0.1473 /therm

Next 30 therms $0.0579/therm

Over 50 therms  $0.0519/therm

Gas supply charge  $0.43/therm

Calculation:

For using $x$ therms in a month, the rate of distribution charge $0.1473 /therm applied for inItial 20 therms.

The rate of distribution charge $0.0579/therm applied for next 30 therms.

The rate of distribution charge $0.0519/therm applied for therms exceed 50.

In addition monthly customer charge $8.40 and gas charge $0.43/therm also applied.

Therm Distribution charge

  $\{\begin{array}{l}\$0.1473xif0\le x\le 20\\ 20\times \$0.1473+\left(x-20\right)\$0.0579if20\le x<50\\ 20\times \$0.1473+30\times \$0.0579+\$0.519\left(x-50\right)ifx>50\end{array}$

Therefore, the change in one month for using $x$ therms is

  $\begin{array}{l}C\left(x\right)=\{\begin{array}{l}\$0.1473x+\$8.40+\$0.43xif0\le x\le 20\\ 20\times \$0.1473+\left(x-20\right)\$0.0579+\$8.40+\$0.43xif20\le x<50\\ 20\times \$0.1473+30\times \$0.0579+\$0.519\left(x-50\right)+\$8.40+\$0.43xifx>50\end{array}\\ C\left(x\right)=\{\begin{array}{l}\$0.5773x+\$8.40if0\le x\le 20\\ \$0.4879x+\$10.188if20\le x<50\\ \$0.4819x+\$10.488ifx>50\end{array}\end{array}$

(d)

To determine

To sketch: the graph of the function which is found in part (c).

Explanation of Solution

Given information:

Monthly customer charge $8.40.

Per therm service charge

1^st 20 therms $0.1473 /therm

Next 30 therms $0.0579/therm

Over 50 therms  $0.0519/therm

Gas supply charge  $0.43/therm

Calculation:

The graph of the function $C\left(x\right)$ is shown below: