Chapter 16, Problem 27P

To determine

The tunnel which should be constructed.

Answer to Problem 27P

The half capacity tunnel should be constructed as the total effective cost for half capacity tunnel is $\$477022$ which is less than the total effective cost for full capacity tunnel of $\$532970$.

Explanation of Solution

Given:

The table for given data is shown below.

Year	Particulars	Full capacity tunnel	Half capacity tunnel
0	Capital cost	$\$500000$	$\$300000$
Beginning of ${20}^{\text{th}}$ year	Capital cost for $2^{\text{nd}}$ half capacity tunnel		$\$300000$
10	Repair	$\$20000$	$\$1600$
20	Repair	$\$20000$	$\$1600$
30	Repair	$\$20000$	$\$3200$
40	Repair	$\$20000$	$\$3200$

Table (1).

Concept used:

Write the expression for total effective cost.

$A=\left[\begin{array}{l}C+C_{20}\left[\frac{1}{{\left(1+i\right)}^{19}}\right]+R_{10}\left[\frac{1}{{\left(1+i\right)}^{10}}\right]+R_{20}\left[\frac{1}{{\left(1+i\right)}^{20}}\right]\\ +R_{30}\left[\frac{1}{{\left(1+i\right)}^{30}}\right]+R_{40}\left[\frac{1}{{\left(1+i\right)}^{40}}\right]\end{array}\right]$ ...... (I)

Here, capital cost is $C$, capital cost at the beginning of ${20}^{\text{th}}$ year is $C_{20}$, repair cost at end of $10$ years is $R_{10}$, repair cost at end of $20$ years is $R_{20}$, repair cost at end of $30$ years is $R_{30}$ and repair cost at end of $40$ years is $R_{40}$.

Calculations:

Calculate the total effective cost for full capacity tunnel.

Substitute $\$500000$ for $C$, $\$20000$ for $R_{10}$, $\$20000$ for $R_{20}$, $\$20000$ for $R_{30}$, $\$20000$ for $R_{40}$, $4\%$ for $i$ and $20$ for $n$ in Equation (I).

$\begin{array}{c}A=\left[\begin{array}{l}\$500000+\$20000\left[\frac{1}{{\left(1+0.04\right)}^{10}}\right]+\$20000\left[\frac{1}{{\left(1+0.04\right)}^{20}}\right]\\ +\$20000\left[\frac{1}{{\left(1+0.04\right)}^{30}}\right]+\$20000\left[\frac{1}{{\left(1+0.04\right)}^{40}}\right]\end{array}\right]\\ =\left[\begin{array}{l}\$500000+\$20000\left(0.6755\right)+\$20000\left(0.4564\right)\\ +\$20000\left(0.3083\right)+\$20000\left(0.2083\right)\end{array}\right]\\ =\$532970\end{array}$

Calculate the total effective cost for half capacity tunnel.

Substitute $\$300000$ for $C$, $\$300000$ for $C_{20}$, $\$1600$ for $R_{10}$, $\$1600$ for $R_{20}$, $\$3200$ for $R_{30}$, $\$3200$ for $R_{40}$, $4\%$ for $i$ and $20$ for $n$ in Equation (I).

$\begin{array}{c}A=\left[\begin{array}{l}\$300000+\$300000\left[\frac{1}{{\left(1+0.04\right)}^{19}}\right]+\$16000\left[\frac{1}{{\left(1+0.04\right)}^{10}}\right]\\ +\$16000\left[\frac{1}{{\left(1+0.04\right)}^{20}}\right]+\$32000\left[\frac{1}{{\left(1+0.04\right)}^{30}}\right]+\$32000\left[\frac{1}{{\left(1+0.04\right)}^{40}}\right]\end{array}\right]\\ \left[\begin{array}{l}\$300000+\$300000\left(0.4746\right)+\$16000\left(0.6755\right)\\ +\$16000\left(0.4564\right)+\$32000\left(0.3083\right)+\$32000\left(0.2083\right)\end{array}\right]\\ =\$477022\end{array}$.

Conclusion:

Thus, the half capacity tunnel should be constructed as the total effective cost for half capacity tunnel is $\$477022$ which is less than the total effective cost for full capacity tunnel of $\$532970$.