Question

a small company heats its building using a gas furnace. it expects to spend $9,900 on natural gas this first year. the cost of the natural gas is expected to increase by 10% per year every subsequent year. there is also the maintenance cost on the gas furnace which is expected to be $851.20 in the first year and to increase by 12% per year every subsequent year. the planning horizon is 13 years and the company uses an annual interest rate of 15% for economic analysis of its investments. (a) the projected annual cost of natural gas in year 13 is $\square$ (round to nearest dollar) (b) the projected annual maintenance cost of the furnace in year 13 is $\square$ (round to nearest dollar) (c) the total present worth of operating and maintaining the furnace over the 13 years is $\square$ (round to nearest dollar)

Explanation:

Part (a)

Step1: Identify the formula for compound growth

The formula for compound growth is $F = P(1 + r)^{n - 1}$, where $P$ is the initial amount, $r$ is the growth rate, and $n$ is the year number. For natural gas cost, $P = 9900$, $r = 0.10$, and $n = 13$.

Step2: Calculate the cost in year 13

Substitute the values into the formula: $F = 9900(1 + 0.10)^{13 - 1}$. First, calculate the exponent: $13 - 1 = 12$. Then, $(1.10)^{12}\approx3.138428376721$. Multiply by 9900: $9900\times3.138428376721\approx31070.44$.

Step1: Use the compound growth formula

For maintenance cost, $P = 851.20$, $r = 0.12$, and $n = 13$. The formula is $F = P(1 + r)^{n - 1}$.

Step2: Compute the maintenance cost in year 13

Calculate the exponent: $13 - 1 = 12$. $(1.12)^{12}\approx3.895975992546$. Multiply by 851.20: $851.20\times3.895975992546\approx3316.26$.

Step1: Recall the present worth formula for geometric series

The present worth $P$ of a geometric series with first term $A_1$, growth rate $g$, interest rate $i$, and number of periods $n$ is $P=\frac{A_1[1 - (\frac{1 + g}{1 + i})^n]}{i - g}$ (when $i
eq g$). We have two series: one for natural gas (NG) and one for maintenance (M).

Step2: Calculate present worth of natural gas cost

For NG: $A_{1,NG}=9900$, $g_{NG}=0.10$, $i = 0.15$, $n = 13$. Substitute into the formula: $P_{NG}=\frac{9900[1 - (\frac{1 + 0.10}{1 + 0.15})^{13}]}{0.15 - 0.10}$. Calculate $\frac{1.10}{1.15}\approx0.9565217391$, raise to 13th power: $(0.9565217391)^{13}\approx0.52302806$. Then $1 - 0.52302806 = 0.47697194$. Divide by 0.05: $\frac{0.47697194}{0.05}=9.5394388$. Multiply by 9900: $9900\times9.5394388\approx94440.44$.

Step3: Calculate present worth of maintenance cost

For M: $A_{1,M}=851.20$, $g_{M}=0.12$, $i = 0.15$, $n = 13$. Substitute into the formula: $P_{M}=\frac{851.20[1 - (\frac{1 + 0.12}{1 + 0.15})^{13}]}{0.15 - 0.12}$. Calculate $\frac{1.12}{1.15}\approx0.97391304$, raise to 13th power: $(0.97391304)^{13}\approx0.6304767$. Then $1 - 0.6304767 = 0.3695233$. Divide by 0.03: $\frac{0.3695233}{0.03}\approx12.317443$. Multiply by 851.20: $851.20\times12.317443\approx10484.60$.

Step4: Total present worth

Add the two present worths: $P_{total}=P_{NG}+P_{M}=94440.44 + 10484.60 = 104925.04$.

Answer:

31070

Part (b)