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Q. 50

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Found in: Page 102

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# In March $2011$, Nicor Gas had the following rate schedule for natural gas usage in single-family residences:Monthly customer charge $-13.55$Distribution chargeFirst $20$ therms$-0.1473/therm$Next $30$ therm $-0.0579/therm$Over $50$ therm $-0.0519/therm$Gas supply charge$-0.51/therm$(a) What is the charge for using $40$ therms in a month?(b) What is the charge for using $150$ therms in a month?(c) Develop a model that gives the monthly charge $C$ for $x$ therms of gas.(d) Graph the function found in part (c).

(a) The charge for using $40$ therms in a month is $38.054$

(b) The charge for using $150$ therms in a month is $99.923$

(c) The model that gives the monthly charge $C$ for $x$ therms of gas is $C\left(x\right)=\left\{\begin{array}{l}0.5619x+2.088,x>50\\ 0.5679x+1.788,x\le 50\end{array}\right\$

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

See the step by step solution

## Part (a) Step 1. Given

Monthly customer charge$-13.55$

First $20$ therms$-0.1473/therm$

Next $30$ therms$-0.0579/therm$

Over $50$ therms$-0.0519/therm$

Gas supply charge$-0.51/therm$

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

The distribution charge for first $20$ therms is

$=0.1473×20\phantom{\rule{0ex}{0ex}}=2.946$

The distribution charge for the next $20$ therms is

$=0.0579×20\phantom{\rule{0ex}{0ex}}=1.158$

Gas charge for $40$ therms $=0.51×40$

$=20.4$

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

Charge for using $40$ therms in a month$=13.55+2.946+1.158+20.4\phantom{\rule{0ex}{0ex}}=38.054$

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

Referring to part (a), the distribution charge for using first $20$ therms is $2.946$

The distribution charge for the next $30$ therms $=0.0579×30$

$=1.737$

The distribution charge for next $100$ therms,

$=0.0519×100\phantom{\rule{0ex}{0ex}}=51.9$

Gas charge for $150$ therms $=0.51×150$

$=76.5$

## Part (b) Step 2. Find the total charge

Monthly service charge$=13.55$

Total charge for using $150$ therms is given by,

$=13.55+2.946+1.737+5.19+76.5\phantom{\rule{0ex}{0ex}}=99.923$

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

Monthly charge $=13.55$

Referring to part (a), (b), the distribution charge for $20$ therms is $2.946$ and the distribution charge for next $30$ therm is $1.737$

The distribution charge for next $\left(x-50\right)$ therms is,

$=0.0519\left(x-50\right)\phantom{\rule{0ex}{0ex}}=0.0519x-2.595$

The gas charge for $x$ therms$=0.51x$

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

For $x\ge 50$ the total charge is given by

$C\left(x\right)=2.946+1.737+0.0519x-2.595+0.51x\phantom{\rule{0ex}{0ex}}=0.5619x+2.088$

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

The total charge for $x$ terms if $x<50$ is given by,

$=0.0579\left(x-20\right)\phantom{\rule{0ex}{0ex}}=0.0579x-1.158$

Total charge for $x$ therms$C\left(x\right)=2.946+0.0579x-1.158+0.51x\phantom{\rule{0ex}{0ex}}=0.5679x+1.788$

## Part (d) Step 1. Graph the function in part (c)

Graph the function $C\left(x\right)=\left\{\begin{array}{l}0.5619x+2.088,x>50\\ 0.5679x+1.788,x\le 50\end{array}\right\$