Question

5-37 the larkspur furniture company needs a new grinder. compute the present worth for these mutually exclusive alternatives and identify which you would recommend given i = 6% per year. larkspur uses a 10 - year planning horizon.
alternative a b
initial cost $4500 $5500
annual costs $300 $400
salvage value $500 $0
life 5 years 10 years
contributed by gillian nicholls, southeast missouri state university

Explanation:

Step1: Calculate present - worth of Alternative A

The present - worth formula for an investment with initial cost $P$, annual cost $A$, salvage value $S$, interest rate $i$, and life $n$ is $PW=-P - A(P/A,i,n)+S(P/F,i,n)$. For Alternative A, $P = 4500$, $A = 300$, $S = 500$, $i=0.06$, and $n = 5$. Since the planning horizon is 10 years and the life of Alternative A is 5 years, we need to consider two cycles.
The present - worth factor for an annuity $(P/A,i,n)=\frac{(1 + i)^n-1}{i(1 + i)^n}$ and the present - worth factor for a single payment $(P/F,i,n)=\frac{1}{(1 + i)^n}$.
First, for one 5 - year cycle:
$(P/A,0.06,5)=\frac{(1 + 0.06)^5-1}{0.06(1 + 0.06)^5}=\frac{1.3382255776 - 1}{0.06\times1.3382255776}=\frac{0.3382255776}{0.08029353466}\approx4.21236$.
$(P/F,0.06,5)=\frac{1}{(1 + 0.06)^5}=\frac{1}{1.3382255776}\approx0.747258$.
The present worth of one 5 - year cycle of Alternative A is $PW_1=-4500-300\times4.21236 + 500\times0.747258=-4500-1263.708+373.629=-5390.079$.
For two 5 - year cycles, $PW_A=-5390.079-5390.079\times(P/F,0.06,5)=-5390.079-5390.079\times0.747258=-5390.079-4023.97=-9414.049$.

Step2: Calculate present - worth of Alternative B

For Alternative B, $P = 5500$, $A = 400$, $S = 0$, $i = 0.06$, and $n = 10$.
$(P/A,0.06,10)=\frac{(1 + 0.06)^{10}-1}{0.06(1 + 0.06)^{10}}=\frac{1.7908477 - 1}{0.06\times1.7908477}=\frac{0.7908477}{0.107450862}\approx7.36009$.
$PW_B=-5500-400\times7.36009+0=-5500 - 2944.036=-8444.036$.

Answer:

The present worth of Alternative A is approximately $\$-9414.05$ and the present worth of Alternative B is approximately $\$-8444.04$. Recommend Alternative B since it has a higher (less negative) present worth.