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Q. 50

Expert-verifiedFound in: Page 102

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 charge

First $20$ 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:

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$-\$0.51/therm$

The distribution charge for first $20$ therms is

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

The distribution charge for the next $20$ therms is

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

Gas charge for $40$ therms $=0.51\times 40$

$=\$20.4$

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

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

The distribution charge for the next $30$ therms $=0.0579\times 30$

$=\$1.737$

The distribution charge for next $100$ therms,

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

Gas charge for $150$ therms $=0.51\times 150$

$=\$76.5$

Monthly service charge$=\$13.55$

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

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

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 $(x-50)$ therms is,

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

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

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

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

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

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

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

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

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