Question

In March $2011$, Peoples Energy had the following rate schedule for natural gas usage in single- family residences:

Monthly service charge- $\backslash (18.95$

Per therm service charge

$1st50$therms- $\backslash )0.33372/therm$

Over $50$therms- $\backslash (0.12360/therm$

Gas charge- $\backslash )0.5038/therm$

(a) What is the charge for using $50$therms in a month?

(b) What is the charge for using $500$therms in a month?

(c) Develop a model that relates the monthly charge $C$for $x$therms of gas.

(d) Graph the function found in part (c)

Short answer

(a) The charge for using $50$therms in a month is $\$60.826$

(b) The charge for using $500$therms in a month is $\$343.156$

(c) The model that relates the monthly charge $C$for $x$therms is $C\left(x\right)=\left\{\begin{array}{l}0.6274x+29.456.x<50\\ 0.83752x+18.95,x>50\end{array}\right.$

(d) The graph of the function found in part (c) is

Step-by-step solution

Part (a) Step 1. Given

Monthly service charge- $\$18.95$

Per therm service charge

$1st50$therms-$\$0.33372/therm$

Over $50$therms-$\$0.12360/therm$

Gas charge-$\$0.5038/therm$

Part (a) Step 2. Find the service charge and gas charge

The service charge for the first $50$therms$=\$0.33372/therm$

The total service charge for the first $50$therms $=0.33372\times 50$

$=\$16.686$

Gas charge for $50$therms $=\$0.5038/therm$

$$\begin{gathered} =0.5038\times 50 \\ =\$25.19 \end{gathered}$$

Part (a) Step 3. Add the terms found above.

Monthly service charge$=\$18.95$

Charge for using $50$therms in a month$=16.686+25.19+18.95$

$=\$60.826$

Part (b) Step 1. Find the service charge and gas charge

Referring to part (a), the service charge for using first $50$therms is $\$16.686$

The service charge for next $450$therms is given by,

$$\begin{gathered} =\$0.12360/therm \\ =0.12360\times 450 \\ =\$55.62 \end{gathered}$$

Gas charge$=0.5038/therm$

$$\begin{gathered} =0.5038\times 500 \\ =\$251.9 \end{gathered}$$

Part (b) Step 2. Find the total charge

Monthly service charge $=\$18.95$

Total charge for using $500$therms$=16.686+55.62+251.9+18.95$

$=\$343.156$

Part (c) Step 1. To find the charges for therms

Monthly charge$=\$18.95$

Referring to part (a) and (b), the service charge for first $50$therms is $\$16.686$

And the service charge for $(x-50)$therms is $=0.12360(x-50)$

$=0.12360x-6.18$

Gas charge for $x$ therms$=0.5038x$

Part (c) Step 2. Find model C for x therms for x>50

For $x<50,$the total charge is given by

$$\begin{gathered} C\left(x\right)=18.95+16.686+0.12360x-6.18+0.5038x \\ =29.456+0.6274x \end{gathered}$$

Part (c) Step 3. Find the model for x>50

The total charge for $x$terms is given by,

$$\begin{gathered} C\left(x\right)=18.95+(0.33372+0.5038)x \\ =18.95+0.83752x \end{gathered}$$

Part (d) Step 4. Graph the function in part (c)

Graph the function $C\left(x\right)=\left\{\begin{array}{l}0.6274x+29.456.x<50\\ 0.83752x+18.95,x>50\end{array}\right.$