Chapter 9, Problem 75P

To determine

(a)

The resale value of alternative B to provide equal economical transportation.

Answer to Problem 75P

The resale value of alternative B is $-\$19.48$ to provide equal economical transportation.

Explanation of Solution

Given:

For alternative A:

The initial cost is $\$8900$.

The resale value is $\$1700$.

The operation cost per kilometer is $\$0.09$.

For alternative B:

The initial cost is $\$8900$.

The operation cost per kilometer is $\$0.075$.

The distance travelled by car is $12000\text{km}$.

The interest rate is $8\%$.

The time period is $3\text{years}$.

Concept used:

Write the expression for economical annual cost.

$EAC=P\left[\frac{i{\left(1+i\right)}^n}{{\left(1+i\right)}^n-1}\right]-S\left[\frac{i}{{\left(1+i\right)}^n-1}\right]+R\times D$ ...... (I)

Here, the initial cost is $P$, the resale value is $S$, the operation cost per kilometer is $R$, the distance travelled by car is $D$, the interest rate is $i$ and time period is $n$.

Calculations:

Calculate the economical annual cost for alternative A.

Substitute $\$8900$ for $P$, $\$1700$ for $S$, $0.09$ for $R$, $12000$ for $D$, $8\%$ for $i$ and $3$ for $n$ in Equation (I).

$\begin{array}{c}EA{C}_A=\$8900\left[\frac{0.08{\left(1+0.08\right)}^3}{{\left(1+0.08\right)}^3-1}\right]-\$1700\left[\frac{0.08}{{\left(1+0.08\right)}^3-1}\right]+\left(\$0.09\times 12000\right)\\ =\$8900\left(0.388\right)-\$1700\left(0.3080\right)+\$1080\\ =\$3453.5-\$523.6+\$1080\\ =\$4010\end{array}$

Calculate the economical annual cost for alternative B.

Substitute $\$8000$ for $P$, $\$0.075$ for $R$, $12000$ for $D$, $8\%$ for $i$ and $3$ for $n$ in Equation (I).

$\begin{array}{c}EA{C}_B=\$8000\left[\frac{0.08{\left(1+0.08\right)}^3}{{\left(1+0.08\right)}^3-1}\right]-S\left[\frac{0.08}{{\left(1+0.08\right)}^3-1}\right]+\left(\$0.075\times 12000\right)\\ =\$8000\left(0.388\right)-S\left(0.3080\right)+\$900\\ =\$3104-0.3080S+\$900\\ =\$4004-0.3080S\end{array}$

Equate the economical annual cost for alternative A with the economical annual cost for alternative B.

$\begin{array}{l}EA{C}_A=EA{C}_B\\ \$4010=\$4004-0.3080S\\ -\$6=0.3080S\\ S=-\$19.48\end{array}$.

Conclusion:

Thus, the resale value of alternative B is $-\$19.48$ to provide equal economical transportation.

To determine

(b)

The resale value of alternative B to provide equal economical transportation.

Answer to Problem 75P

The resale value of alternative B is $\$272.72$ to provide equal economical transportation.

Explanation of Solution

Given:

For alternative A:

The initial cost is $\$8900$.

The resale value is $\$1700$.

The operation cost per kilometer is $\$0.09$.

For alternative B:

The initial cost is $\$8900$.

The operation cost per kilometer is $\$0.08$.

The distance travelled by car is $9000\text{km}$.

The interest rate is $8\%$.

The time period is $3\text{years}$.

Calculations:

Calculate the economical annual cost for alternative A.

Substitute $\$8900$ for $P$, $\$1700$ for $S$, $\$0.09$ for $R$, $9000$ for $D$, $8\%$ for $i$ and $3$ for $n$ in Equation (I).

$\begin{array}{c}EA{C}_A=\$8900\left[\frac{0.08{\left(1+0.08\right)}^3}{{\left(1+0.08\right)}^3-1}\right]-\$1700\left[\frac{0.08}{{\left(1+0.08\right)}^3-1}\right]+\left(\$0.09\times 9000\right)\\ =\$8900\left(0.388\right)-\$1700\left(0.3080\right)+\$810\\ =\$3453.5-\$523.6+\$810\\ =\$3740\end{array}$

Calculate the economical annual cost for alternative B.

Substitute $\$8000$ for $P$, $\$0.08$ for $R$, $9000$ for $D$, $8\%$ for $i$ and $3$ for $n$ in Equation (I).

$\begin{array}{c}EA{C}_B=\$8000\left[\frac{0.08{\left(1+0.08\right)}^3}{{\left(1+0.08\right)}^3-1}\right]-S\left[\frac{0.08}{{\left(1+0.08\right)}^3-1}\right]+\left(\$0.08\times 9000\right)\\ =\$8000\left(0.388\right)-S\left(0.3080\right)+\$720\\ =\$3104-0.3080S+\$720\\ =\$3824-0.3080S\end{array}$

Equate the economical annual cost for alternative A with the economical annual cost for alternative B.

$\begin{array}{l}EA{C}_A=EA{C}_B\\ \$3740=\$3824-0.3080S\\ \$84=0.3080S\\ S=\$272.72\end{array}$.

Conclusion:

Thus, the resale value of alternative B is $\$272.72$ to provide equal economical transportation.

To determine

(c)

The resale value of alternative B to provide equal economical transportation.

Answer to Problem 75P

The resale value of alternative B is $\$251.2$ to provide equal economical transportation.

Explanation of Solution

Given:

For alternative A:

The initial cost is $\$8900$.

The resale value is $\$1700$.

The operation cost per kilometer is $\$0.09$.

For alternative B:

The initial cost is $\$8900$.

The operation cost per kilometer is $\$0.08$.

The distance travelled by car is $12000\text{km}$.

The interest rate is $6\%$.

The time period is $3\text{years}$.

Calculations:

Calculate the economical annual cost for alternative A.

Substitute $\$8900$ for $P$, $\$1700$ for $S$, $\$0.09$ for $R$, $12000$ for $D$, $6\%$ for $i$ and $3$ for $n$ in Equation (I).

$\begin{array}{c}EA{C}_A=\$8900\left[\frac{0.06{\left(1+0.06\right)}^3}{{\left(1+0.06\right)}^3-1}\right]-\$1700\left[\frac{0.06}{{\left(1+0.06\right)}^3-1}\right]+\left(\$0.09\times 12000\right)\\ =\$8900\left(0.374\right)-\$1700\left(0.3141\right)+\$1080\\ =\$3328.6-\$533.97+\$1080\\ =\$3874.63\end{array}$

Calculate the economical annual cost for alternative B.

Substitute $\$8000$ for $P$, $\$0.08$ for $R$, $12000$ for $D$, $6\%$ for $i$ and $3$ for $n$ in Equation (I).

$\begin{array}{c}EA{C}_B=\$8000\left[\frac{0.06{\left(1+0.06\right)}^3}{{\left(1+0.06\right)}^3-1}\right]-S\left[\frac{0.06}{{\left(1+0.06\right)}^3-1}\right]+\$0.08\times 12000\\ =\$8000\left(0.374\right)-S\left(0.3141\right)+\$960\\ =\$2992-0.3141S+\$960\\ =\$3952-0.3141S\end{array}$

Equate the economical annual cost for alternative A with the economical annual cost for alternative B.

$\begin{array}{l}EA{C}_A=EA{C}_B\\ \$3874.63=\$3952-0.3141S\\ \$77.37=0.3080S\\ S=\$251.2\end{array}$.

Conclusion:

Thus, the resale value of alternative B is $\$251.2$ to provide equal economical transportation.

To determine

(d)

The resale value of alternative B to provide equal economical transportation.

Answer to Problem 75P

The resale value of alternative B is $\$202.5$ to provide equal economical transportation.

Explanation of Solution

Given:

For alternative A:

The initial cost is $\$8900$.

The resale value is $\$1700$.

The operation cost per kilometer is $\$0.09$.

For alternative B:

The initial cost is $\$8900$.

The operation cost per kilometer is $\$0.075$.

The distance travelled by car is $9000\text{km}$.

The interest rate is $6\%$.

The time period is $3\text{years}$.

Calculations:

Calculate the economical annual cost for alternative A.

Substitute $\$8900$ for $P$, $\$1700$ for $S$, $\$0.09$ for $R$, $9000$ for $D$, $6\%$ for $i$ and $3$ for $n$ in Equation (I).

$\begin{array}{c}EA{C}_A=\$8900\left[\frac{0.06{\left(1+0.06\right)}^3}{{\left(1+0.06\right)}^3-1}\right]-\$1700\left[\frac{0.06}{{\left(1+0.06\right)}^3-1}\right]+\left(\$0.09\times 9000\right)\\ =\$8900\left(0.374\right)-\$1700\left(0.3141\right)+\$810\\ =\$3328.6-\$533.97+\$810\\ =\$3604.63\end{array}$

Calculate the economical annual cost for alternative B.

Substitute $\$8000$ for $P$, $0.075$ for $R$, $9000$ for $D$, $6\%$ for $i$ and $3$ for $n$ in Equation (I).

$\begin{array}{c}EA{C}_B=\$8000\left[\frac{0.06{\left(1+0.06\right)}^3}{{\left(1+0.06\right)}^3-1}\right]-S\left[\frac{0.06}{{\left(1+0.06\right)}^3-1}\right]+\left(\$0.075\times 9000\right)\\ =\$8000\left(0.374\right)-S\left(0.3141\right)+\$675\\ =\$2992-0.3141S+\$675\\ =\$3667-0.3141S\end{array}$

Equate the economical annual cost for alternative A with the economical annual cost for alternative B.

$\begin{array}{l}EA{C}_A=EA{C}_B\\ \$3604.63=\$3667-0.3141S\\ \$62.37=0.3080S\\ S=\$202.5\end{array}$.

Conclusion:

Thus, the resale value of alternative B is $\$202.5$ to provide equal economical transportation.