Chapter 2, Problem 13P

To determine

(a)

The monthly bill for the shop, the marginal cost per kilowatt hour and the average costs per kilowatt hour.

Answer to Problem 13P

The monthly bill of the shop is $22145¢$.

The marginal cost of the shop is $7.61¢$.

The average cost of the shop is $7.91¢$.

Explanation of Solution

Given:

Load on the machine is $30\text{hp}$.

Electricity rates are as follows:

$\text{First}50\text{kWh}\text{at}12.6\text{per}\text{kWh}$

$\text{Next}50\text{kWh}\text{at}10.6\text{per}\text{kWh}$

$\text{Next}150\text{kWh}\text{at}9.0\text{per}\text{kWh}$

$\text{Over}250\text{kWh}\text{at}7.7\text{per}\text{kWh}$.

Calculation:

Write the expression to calculate the average cost.

$\text{Average}\text{cost}=\frac{\left(\text{Total}\text{consumption}\right)\times \left(\text{rate}\right)}{\text{Total}\text{consumption}}$ ...... (I)

Substitute the value in the above expression.

$\begin{array}{c}\text{Average}\text{cost}=\frac{\left[\begin{array}{l}\left(50\text{kWh}\times 12.6¢\right)+\left(50\text{kWh}\times 10.6¢\right)\\ +\left(150\text{kWh}\times 9¢\right)+\left(2550\text{kWh}\times 7.7¢\right)\end{array}\right]}{\left(50\text{kWh}+50\text{kWh}+150\text{kWh}+2550\text{kWh}\right)}\\ =\frac{22145\text{kWh}¢}{2800\text{kWh}}\\ =7.91¢\end{array}$

Write the expression to calculate the monthly bill of the shop.

$\text{Monthly}\text{bill}=\left(\text{Total}\text{consumption}\right)\times \text{rate}$ ..... (II)

Substitute the value in the above expression.

$\begin{array}{c}\text{Monthly}\text{bill}=\left[\begin{array}{l}\left(50\text{kWh}\times 12.6¢\right)+\left(50\text{kWh}\times 10.6¢\right)+\\ \left(150\text{kWh}\times 9¢\right)+\left(2550\text{kWh}\times 7.7¢\right)\end{array}\right]\\ =22145¢\end{array}$

Write the expression to calculate the marginal cost.

$\text{Marginal}\text{cost}=\left(\frac{\text{Total}{\text{cost}}_{\left(\text{n}\right)}-\text{Total}{\text{cost}}_{\left(\text{n-1}\right)}}{{\text{Consumption}}_{\left(\text{n}\right)}-{\text{Consumption}}_{\left(\text{n-1}\right)}}\right)$ ...... (III)

Substitute, $19635\text{kWh}¢$ for $\text{total}{\text{cost}}_{\left(\text{n}\right)}$, $1350\text{kWh}¢$ for $\text{total}{\text{cost}}_{\left(\text{n-1}\right)}$, $2550\text{kWh}$ for ${\text{consumption}}_{\left(\text{n}\right)}$ and $150\text{kWh}$ for ${\text{consumption}}_{\left(\text{n-1}\right)}$ in Equation (III)

$\begin{array}{c}\text{Marginal}\text{cost}=\frac{19635\text{kWh}¢-1350\text{kWh}¢}{2550\text{kWh}-150\text{kWh}}\\ =\frac{18285\text{kWh¢}}{2400\text{kWh}}\\ =7.61¢\end{array}$.

Conclusion:

The monthly bill of the shop is $22145¢$.

The marginal cost of the shop is $7.61¢$.

The average cost of the shop is $7.91¢$.

To determine

(b)

The cost of additional energy and the cost per kilowatt hour.

Answer to Problem 13P

The cost for additional energy is $9240¢$.

The cost per kilowatt hour is $3.3¢$.

Explanation of Solution

Given:

Additional energy requirement is $1200\text{kWh}$.

Calculation:

Write the expression to calculate the additional energy cost

$\text{Cost}=\left(\text{additional}\text{energy}\right)\times \left(\text{rate}\right)$ ...... (I)

Substitute, $1200\text{kWh}$ for additional energy and $7.7¢$ for rate in Equation (I).

$\begin{array}{c}\text{Cost}=1200\text{kWh}\times 7.7¢\\ =9240\text{¢}\end{array}$

Write the expression to calculate the cost of additional energy per kilowatt-hour.

$\text{Cost}/\text{kilowatt-hour}=\frac{\text{Cost}}{\text{Total}\text{consumption}}$ ...... (II)

Substitute, $9240\text{kWh¢}$ for cost and $2880\text{kWh}$ for total consumption in Equation (II)

$\begin{array}{c}\text{Cost}/\text{kilowatt-hour}=\frac{\text{9240}\text{kWh}¢}{\text{2800}\text{kWh}}\\ =3.3¢\end{array}$.

Conclusion:

The cost for additional energy is $9240¢$.

The cost per kilowatt hour is $3.3¢$.

To determine

(c)

Cost for installing new machines and the cost per kilowatt hour.

Answer to Problem 13P

The cost for installing new machines is $770$.

The cost per kilowatt hour is $0.275¢$.

Explanation of Solution

Given:

Power consumption by new machines is $100\text{kWh}$.

Calculation:

Write the expression to calculate cost for installing new machines.

$\text{Cost}=\text{Power}\times \text{Rate}$ ...... (I)

Substitute, $100\text{kWh}$ for power and $7.7¢$ for rate in Equation (I)

$\begin{array}{c}\text{Cost}=100\text{kWh}\times 7.7¢\\ =770\text{kWh}¢\end{array}$

Write the expression to calculate the cost of additional energy per kilowatt-hour.

$\text{Cost}/\text{kilowatt-hour}=\frac{\text{Cost}}{\text{Total}\text{consumption}}$ ...... (II)

Substitute, $9240\text{kWh¢}$ for cost and $2880\text{kWh}$ for total consumption in Equation (II)

$\begin{array}{c}\text{Cost}/\text{kilowatt-hour}=\frac{770\text{kWh}¢}{\text{2800}\text{kWh}}\\ =0.275¢\end{array}$.

Conclusion:

The cost for installing new machines is $770$.

The cost per kilowatt hour is $0.275¢$.