Chapter D, Problem 94P

To determine

The annual revenue of bid $A$.

Answer to Problem 94P

The correct answer is option $\left(c\right)0.3$.

Explanation of Solution

Concept used:

Write the expression for present worth of cost.

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

Here, the overall cost is $C$, annual maintenance is $A_O$, interest rate is $i$ and time period is $n$.

Write the expression for present worth of benefit.

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

Here, the annual benefit is $B$.

Calculations:

Calculate the overall cost of bid $B$.

$\begin{array}{c}C=\$6.2\text{ M}+\$0.03\text{ M}+\$0.7\text{ M}+\$0.01\text{ M}\\ =\$\text{6}\text{.94 M}\end{array}$

Calculate the present worth of cost of bid $B$.

Substitute $\$6.94\text{ M}$ for $C$, $\$0.75\text{ M}$ for $A_O$, $5\%$ for $i$ and $30$ for $n$ in Equation (I).

$\begin{array}{c}P{W}_C=-\$6.94\text{ M}+\$0.075\text{ M}\left[\frac{{\left(1+0.05\right)}^{30}-1}{0.05{\left(1+0.05\right)}^{30}}\right]\\ =-\$6.94\text{ M}+\$0.075\text{ M}\left(15.3724\right)\\ =-\$5.87\text{ M}\end{array}$

Calculate the annual benefit of bid $B$.

$\begin{array}{c}B=\$0.4\text{ M}+\$0.25\text{ M}\\ =\$0.\text{65 M}\end{array}$

Calculate the present worth of benefit of bid $B$.

Substitute $\$0.65\text{ M}$ for $B$, $5\%$ for $i$ and $30$ for $n$ in Equation (II).

$\begin{array}{c}P{W}_B=\$0.65\text{ M}\left[\frac{{\left(1+0.05\right)}^{30}-1}{0.05{\left(1+0.05\right)}^{30}}\right]\\ =\$0.65\text{ M}\left[\frac{{\left(1.05\right)}^{30}-1}{0.05{\left(1.05\right)}^{30}}\right]\\ =\$0.65\text{ M}\left(15.3724\right)\\ =\$9.992\text{ M}\end{array}$

Calculate the benefit cost ratio of bid $B$.

$\begin{array}{c}\text{B-C ratio}=\frac{P{W}_B}{P{W}_C}\\ =\frac{\$9.992\text{ M}}{\$5.87\text{ M}}\\ =1.702\end{array}$

Calculate the overall cost of bid $A$.

$\begin{array}{c}C=\$4.2\text{ M}+\$0.6\text{ M}+\$0.35\text{ M}+\$0.6\text{ M}\\ =\$5.\text{75 M}\end{array}$

Calculate the present worth of cost of bid $A$.

Substitute $\$5.75\text{ M}$ for $C$, $\$0.05\text{ M}$ for $A_O$, $5\%$ for $i$ and $30$ for $n$ in Equation (I).

$\begin{array}{c}P{W}_C=-\$5.75\text{ M}+\$0.05\text{ M}\left[\frac{{\left(1+0.05\right)}^{30}-1}{0.05{\left(1+0.05\right)}^{30}}\right]\\ =-\$5.75\text{ M}+\$0.05\text{ M}\left(15.3724\right)\\ =-\$4.98\text{ M}\end{array}$

Calculate the annual benefit of bid $A$.

$B=X+\$0.25\text{ M}$

Calculate the present worth of benefit of bid $A$.

Substitute $\left(X+\$0.25\text{ M}\right)$ for $B$, $5\%$ for $i$ and $30$ for $n$ in Equation (II).

$\begin{array}{c}P{W}_B=\left(X+\$0.25\text{ M}\right)\left[\frac{{\left(1+0.05\right)}^{30}-1}{0.05{\left(1+0.05\right)}^{30}}\right]\\ =\left(X+\$0.25\text{ M}\right)\left[\frac{{\left(1.05\right)}^{30}-1}{0.05{\left(1.05\right)}^{30}}\right]\\ =\left(X+\$0.25\text{ M}\right)\left(15.3724\right)\\ =15.3724X+\$3.843\text{ M}\end{array}$

Calculate the benefit cost ratio of bid $A$.

$\begin{array}{c}\text{B-C ratio}=\frac{15.3724X+\$3.843\text{M}}{\$4.98\text{ M}}\\ =3.087X+0.7717\end{array}$

For both options to be equivalent, the benefit cost ratio of both bids should be Equal.

$\begin{array}{l}1.702=3.087X+0.7717\\ X=\$0.301\text{ M}\end{array}$

The annual revenue of bid $A$ is $\$0.301\text{ M}$.

Conclusion:

Thus, the correct answer is option $\left(c\right)0.3$.