Chapter 9, Problem 32P

To determine

The option which should be selected.

Answer to Problem 32P

The option C the building with height of $10$ stories should be selected.

Explanation of Solution

Given:

Cost of land is $\$100000$ and the interest rate is $8\%$.

Concept used:

Write the expression for present worth of benefits.

$P{W}_B=A\left[\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right]$ ...... (I)

Here, the annual payment is $A$, the interest rate is $i$ and time period is $n$.

Write the expression for present worth of costs.

$P{W}_C=C+L-S\left[\frac{1}{{\left(1+i\right)}^n}\right]$ ...... (II)

Here, the cost of building is $C$, the cost of land is $L$ and the salvage value after $20$ years is $S$.

Write the expression for benefit cost ratio.

$\text{B-C ratio}=\frac{P{W}_B}{P{W}_C}$ ...... (III)

Calculations:

Consider option A the building with height of $2$ stories.

Calculate the present worth of benefits.

Substitute $\$70000$ for $A$, $8\%$ for $i$ and $20$ for $n$ in Equation (I).

$\begin{array}{c}P{W}_B=\$70000\left[\frac{{\left(1+0.08\right)}^{20}-1}{0.08{\left(1+0.08\right)}^{20}}\right]\\ =\$700000\left[\frac{{\left(1.08\right)}^{20}-1}{0.08{\left(1.08\right)}^{20}}\right]\\ =\$70000\left(9.8181\right)\\ =\$687267\end{array}$

Calculate the present worth of costs.

Substitute $\$400000$ for $C$, $\$100000$ for $L$, $\$200000$ for $S$, $8\%$ for $i$ and $20$ for $n$ in Equation (II).

$\begin{array}{c}P{W}_C=\$400000+\$100000-\$200000\left[\frac{1}{{\left(1+0.08\right)}^{20}}\right]\\ =\$500000-\$200000\left(0.2145\right)\\ =\$500000-\$42900\\ =\$457100\end{array}$

Calculate the benefit cost ratio.

Substitute $\$457100$ for $P{W}_C$ and $\$687267$ for $P{W}_B$ in Equation (III).

$\begin{array}{c}\text{B-C ratio}=\frac{\$687267}{\$457100}\\ =1.5\end{array}$

Consider option B the building with height of $5$ stories.

Calculate the present worth of benefits.

Substitute $\$105000$ for $A$, $8\%$ for $i$ and $20$ for $n$ in Equation (I).

$\begin{array}{c}P{W}_B=\$105000\left[\frac{{\left(1+0.08\right)}^{20}-1}{0.08{\left(1+0.08\right)}^{20}}\right]\\ =\$105000\left[\frac{{\left(1.08\right)}^{20}-1}{0.08{\left(1.08\right)}^{20}}\right]\\ =\$105000\left(9.8181\right)\\ =\$1030901\end{array}$

Calculate the present worth of costs.

Substitute $\$800000$ for $C$, $\$100000$ for $L$, $\$300000$ for $S$, $8\%$ for $i$ and $20$ for $n$ in Equation (II).

$\begin{array}{c}P{W}_C=\$800000+\$100000-\$300000\left[\frac{1}{{\left(1+0.08\right)}^{20}}\right]\\ =\$900000-\$300000\left(0.2145\right)\\ =\$900000-\$64350\\ =\$8356500\end{array}$

Calculate the benefit cost ratio.

Substitute $\$8356500$ for $P{W}_C$ and $\$1030901$ for $P{W}_B$ in Equation (III).

$\begin{array}{c}\text{B-C ratio}=\frac{\$1030901}{\$8356500}\\ =1.23\end{array}$

Consider option C the building with height of $10$ stories.

Calculate the present worth of benefits.

Substitute $\$256000$ for $A$, $8\%$ for $i$ and $20$ for $n$ in Equation (I).

$\begin{array}{c}P{W}_B=\$256000\left[\frac{{\left(1+0.08\right)}^{20}-1}{0.08{\left(1+0.08\right)}^{20}}\right]\\ =\$256000\left[\frac{{\left(1.08\right)}^{20}-1}{0.08{\left(1.08\right)}^{20}}\right]\\ =\$256000\left(9.8181\right)\\ =\$2513434\end{array}$

Calculate the present worth of costs.

Substitute $\$800000$ for $C$, $\$100000$ for $L$, $\$300000$ for $S$, $8\%$ for $i$ and $20$ for $n$ in Equation (II).

$\begin{array}{c}P{W}_C=\$2100000+\$100000-\$400000\left[\frac{1}{{\left(1+0.08\right)}^{20}}\right]\\ =\$2100000-\$400000\left(0.2145\right)\\ =\$2200000-\$85800\\ =\$2114200\end{array}$

Calculate the benefit cost ratio.

Substitute $\$2114200$ for $P{W}_C$ and $\$2513434$ for $P{W}_B$ in Equation (III).

$\begin{array}{c}\text{B-C ratio}=\frac{\$2513434}{\$2114200}\\ =1.19\end{array}$

Consider incremental analysis for option B-A.

Calculate the value of A.

$\begin{array}{c}A=\$105000-\$70000\\ =\$35000\end{array}$

Calculate the present worth of benefits.

Substitute $\$35000$ for $A$, $8\%$ for $i$ and $20$ for $n$ in Equation (I).

$\begin{array}{c}P{W}_B=\$35000\left[\frac{{\left(1+0.08\right)}^{20}-1}{0.08{\left(1+0.08\right)}^{20}}\right]\\ =\$350000\left[\frac{{\left(1.08\right)}^{20}-1}{0.08{\left(1.08\right)}^{20}}\right]\\ =\$35000\left(9.8181\right)\\ =\$343634\end{array}$

Calculate the value of $C$.

$\begin{array}{c}C=\$800000-\$400000\\ =\$400000\end{array}$

Calculate the value of $L$.

$\begin{array}{c}L=\$100000-\$100000\\ =0\end{array}$

Calculate the value of $S$.

$\begin{array}{c}S=\$300000-\$200000\\ =\$100000\end{array}$

Calculate the present worth of costs.

Substitute for for for for and for in Equation (II).

Calculate the benefit cost ratio.

Substitute for and for in Equation (III).

The benefit cost ratio is less than so option A is more preferable to B.

Consider incremental analysis for option C-A.

Calculate the present worth of benefits.

Calculate the value of A.

Substitute for for and for in Equation (I).

Calculate the value of

Calculate the value of

Calculate the value of

Calculate the present worth of costs.

Substitute for for for for and for in Equation (II).

Calculate the benefit cost ratio.

Substitute for and for in Equation (III).

The benefit cost ratio is more than so option C is more preferable to A.

The option C the building with height of stories should be selected.

Conclusion:

The option C the building with height of stories should be selected.