Chapter 10, Problem 2P

To determine

The Equivalent uniform cost for $10000\text{miles}$, $15000$ miles, and $5000$ miles.

Answer to Problem 2P

The EUAC for $10000\text{miles}$ is, $\$4725.6$.

The EUAC for $15000\text{miles}$ is, $\$6669.1$.

The EUAC for $5000\text{miles}$ is, $\$2782.1$.

Explanation of Solution

Given:

Purchasing cost of used pickup is $\$14000$.

Fuel cost is $\$1500$ per year.

The drop of salvage value after $5$ year is $10\%$ per mile.

The number of miles truck has driven is $10000\text{miles}$.

Rate of interest is $6\%$.

Calculation:

Case I: Truck is driven $10000$ miles.

Calculate the salvage value.

$\text{Salvage}\text{value}=\left(\text{Purchase}\text{price}\right)-\left(0.10\times \left(\text{Miles}\text{driven}\text{in}\text{5}\text{years}\right)\right)$ ...... (I)

Substitute $\$14000$ for Purchase price, $10000\text{miles}$ for Miles driven in $5$ years in Equation (I).

$\begin{array}{c}\text{Salvage}\text{value}=\$14000-\left(0.10\times \left(10000\text{miles}\right)\times \text{5}\right)\\ =\$14000-\$5000\\ =\$9000\end{array}$

Calculate the Equivalent uniform annual cost.

$\text{EUAC}=\left[\begin{array}{l}\text{Yearly}\text{fuel}\text{cost}+\text{Yearly}\text{other}\text{costs}\\ +\text{Purchase}\text{price}\left[\frac{i{\left(1+i\right)}^n}{{\left(1+i\right)}^n-1}\right]\\ -\text{Salvage}\text{value}\left[\frac{i}{{\left(1+i\right)}^n-1}\right]\end{array}\right]$ ...... (II)

Here, the Equivalent uniform annual cost is EUAC, interest rate is i, and number of year is n.

Substitute $\$1500$ for Yearly fuel cost and Yearly other cost, $\$14000$ for Purchase price, $\$9000$ for the Salvage value, $6\%$ for i, and $5$ year for n in Equation (II).

$\begin{array}{c}\text{EUAC}=\left[\begin{array}{l}\$1500+\$1500\\ +\$14000\left[\frac{0.06{\left(1+0.06\right)}^5}{{\left(1+0.06\right)}^5-1}\right]-\\ \$9000\left[\frac{0.06}{{\left(1+0.06\right)}^5-1}\right]\end{array}\right]\\ =\$3000+\left(\$14000\times 0.2373\right)-\left(\$9000\times 0.1774\right)\\ =\$3000+\$3322.2-\$1596.6\\ =\$4725.6\end{array}$

Case II: Truck is driven $15000$ miles.

Calculate the fuel cost.

$\begin{array}{c}\text{Fuel}\text{cost}=\frac{\$1500}{\$10000}\times \$15000\\ =\$2250\end{array}$

Calculate the salvage value.

Substitute $\$14000$ for Purchase price, $15000\text{miles}$ for miles driven in $5$ year in Equation (I).

$\begin{array}{c}\text{Salvage}\text{value}=\$14000-\left(0.10\times \left(15000\text{miles}\right)\times \text{5}\right)\\ =\$14000-\$7500\\ =\$6500\end{array}$

Calculate the Equivalent uniform annual cost.

Substitute $\$2250$ for Yearly fuel cost and Yearly other cost, $\$14000$ for Purchase price, $\$6500$ for the Salvage value, $6\%$ for i, and $5$ year for n in Equation (II).

$\begin{array}{c}\text{EUAC}=\left[\begin{array}{l}\$2250+\$2250\\ +\$14000\left[\frac{0.06{\left(1+0.06\right)}^5}{{\left(1+0.06\right)}^5-1}\right]\\ -\$6500\left[\frac{0.06}{{\left(1+0.06\right)}^5-1}\right]\end{array}\right]\\ =\$4500+\left(\$14000\times 0.2373\right)-\left(\$6500\times 0.1774\right)\\ =\$4500+\$3322.2-\$1153.1\\ =\$6669.1\end{array}$

Case III: Truck is driven $5000$ miles.

Calculate the fuel cost.

$\begin{array}{c}\text{Fuel}\text{cost}=\frac{\$1500}{\$10000}\times \$5000\\ =\$750\end{array}$

Calculate the salvage value.

Substitute $\$14000$ for Purchase cost, $15000\text{miles}$ for miles driven in $5$ year in Equation (I).

$\begin{array}{c}\text{Salvage}\text{value}=\$14000-\left(0.10\times \left(5000\text{miles}\right)\times \text{5}\right)\\ =\$14000-\$2500\\ =\$11500\end{array}$

Calculate the Equivalent uniform annual cost.

Substitute $\$750$ for Yearly fuel cost and Yearly other cost, $\$14000$ for Purchase price, $\$11500$ for the Salvage value, $6\%$ for i, and $5$ year for n in Equation (II).

$\begin{array}{c}\text{EUAC}=\left[\begin{array}{l}\$750+\$750\\ +\$14000\left[\frac{0.06{\left(1+0.06\right)}^5}{{\left(1+0.06\right)}^5-1}\right]-\\ \$11500\left[\frac{0.06}{{\left(1+0.06\right)}^5-1}\right]\end{array}\right]\\ =\$1500+\left(\$14000\times 0.2373\right)-\left(\$11500\times 0.1774\right)\\ =\$1500+\$3322.2-\$2040.1\\ =\$2782.1\end{array}$.

Conclusion:

Therefore, the EUAC for $10000\text{miles}$ is, $\$4725.6$.

The EUAC for $15000\text{miles}$ is, $\$6669.1$.

The EUAC for $5000\text{miles}$ is, $\$2782.1$.