Chapter 1.3, Problem 120AYU

(a)

To determine

A linear equation that represents the relation between the monthly charge $C$ in dollars, and the number $x$ of therms used in a month, if the customers are charged $\$21.82$ plus $64.9$ cents per therm of heat energy.

Answer to Problem 120AYU

Solution:

A linear equation that relates the monthly charge $C$ in dollars, to the number $x$ of therms used in a month is, $C\left(x\right)=0.649x+21.82$

Explanation of Solution

Given Information:

American Illinois supplies natural gas to residential customers for a monthly customer charge of $\$21.82$ plus $64.9$ cents per therm of heat energy.

Explanation:

First, convert $64.9$ cents in dollar.

$1\text{dollar}=100\text{cents}$.

Thus, divide $64.9$ by $100$.

$\Rightarrow \frac{64.9}{100}=0.649$

Therefore, $64.9$ cents is equal to $0.649$ dollar.

Thus, a monthly customer charge of $\$21.82$ is added to $\$0.649$ dollar per kilowatt.

So the linear equation that represents the relation between the monthly charge $C$ in dollars, and the number $x$ therms used in a month is, $C\left(x\right)=0.649x+21.82$.

(b)

To determine

The monthly charge for using $90$ therms, if the customers are charged $\$21.82$ plus $64.9$ cents per therm of heat energy.

Answer to Problem 120AYU

Solution:

The monthly charge for using $90$ therms is $\$80.23$

Explanation of Solution

Given Information:

American Illinois supplies natural gas to residential customers for a monthly customer charge of $\$21.82$ plus $64.9$ cents per therm of heat energy.

Explanation:

From part (a), the linear equation that represents the relation between the monthly charge $C$ in dollars, and the number $x$ therms used in a month is, $C\left(x\right)=0.649x+21.82$.

To find the monthly charge for using $90$ kilowatt hours, plug $x=90$ in a linear equation $C\left(x\right)=0.649x+21.82$.

Thus, $C\left(x\right)=0.649\left(90\right)+21.82$

$\Rightarrow C\left(x\right)=58.41+21.82$

$\Rightarrow C\left(x\right)=\$80.23$

Thus, the monthly charge for using $90$ kilowatt hours is $\$80.23$.

(c)

To determine

The monthly charge for using $150$ therms, if the customers are charged $\$21.82$ plus $64.9$ cents per therm of heat energy.

Answer to Problem 120AYU

Solution:

The monthly charge for using $150$ therms is $\$119.17$

Explanation of Solution

Given Information:

American Illinois supplies natural gas to residential customers for a monthly customer charge of $\$21.82$ plus $64.9$ cents per therm of heat energy.

Explanation:

From part (a), the linear equation that represents the relation between the monthly charge $C$ in dollars, and the number $x$ therms used in a month is, $C\left(x\right)=0.649x+21.82$.

To find the monthly charge for using $150$ kilowatt hours, plug $x=150$ in a linear equation $C\left(x\right)=0.649x+21.82$.

Thus, $C\left(x\right)=0.649\left(150\right)+21.82$

$\Rightarrow C\left(x\right)=97.35+21.82$

$\Rightarrow C\left(x\right)=119.17$

Thus, the monthly charge for using $150$ kilowatt hours is $\$119.17$.

(d)

To determine

To graph: The linear equation $C\left(x\right)=21.82x+0.649$, if the customers are charged $\$21.82$ plus $64.9$ cents per therm of heat energy.

Explanation of Solution

Given Information:

American Illinois supplies natural gas to residential customers for a monthly customer charge of $\$21.82$ plus $64.9$ cents per therm of heat energy.

Graph:

From part (a), the linear equation that represents the relation between the monthly charge $C$ in dollars, and the number $x$ therms used in a month is, $C\left(x\right)=0.649x+21.82$.

By using graphing calculator the graph of the linear equation $C\left(x\right)=0.649x+21.82$ is

Interpretation:

The graph shows the linear equation $C\left(x\right)=0.649x+21.82$.

(e)

To determine

The slope of the line $C\left(x\right)=0.649x+21.82$, if the customers are charged $\$21.82$ plus $64.9$ cents per therm of heat energy.

Answer to Problem 120AYU

Solution:

The slope of line is $0.649$.

Explanation of Solution

Given Information:

American Illinois supplies natural gas to residential customers for a monthly customer charge of $\$21.82$ plus $64.9$ cents per therm of heat energy.

Explanation:

From part (a), the linear equation that represents the relation between the monthly charge $C$ in dollars, and the number $x$ therms used in a month is, $C\left(x\right)=0.649x+21.82$.

The linear equation of line is $C\left(x\right)=0.649x+21.82$.

The equation $y=mx+c$ represent point slope form of the line, where $m$ is slope and $c$ is $y-$ intercept.

Compare equation $C\left(x\right)=0.649x+21.82$ with $y=mx+c$, get $m=0.649$.

Therefore, slope of the line is $0.649$.