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.
PWB=A[(1+i)n−1i(1+i)n] ...... (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.
PWC=C+L−S[1(1+i)n] ...... (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.
B-C ratio=PWBPWC ...... (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).
PWB=$70000[(1+0.08)20−10.08(1+0.08)20]=$700000[(1.08)20−10.08(1.08)20]=$70000(9.8181)=$687267
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).
PWC=$400000+$100000−$200000[1(1+0.08)20]=$500000−$200000(0.2145)=$500000−$42900=$457100
Calculate the benefit cost ratio.
Substitute $457100 for PWC and $687267 for PWB in Equation (III).
B-C ratio=$687267$457100=1.5
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).
PWB=$105000[(1+0.08)20−10.08(1+0.08)20]=$105000[(1.08)20−10.08(1.08)20]=$105000(9.8181)=$1030901
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).
PWC=$800000+$100000−$300000[1(1+0.08)20]=$900000−$300000(0.2145)=$900000−$64350=$8356500
Calculate the benefit cost ratio.
Substitute $8356500 for PWC and $1030901 for PWB in Equation (III).
B-C ratio=$1030901$8356500=1.23
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).
PWB=$256000[(1+0.08)20−10.08(1+0.08)20]=$256000[(1.08)20−10.08(1.08)20]=$256000(9.8181)=$2513434
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).
PWC=$2100000+$100000−$400000[1(1+0.08)20]=$2100000−$400000(0.2145)=$2200000−$85800=$2114200
Calculate the benefit cost ratio.
Substitute $2114200 for PWC and $2513434 for PWB in Equation (III).
B-C ratio=$2513434$2114200=1.19
Consider incremental analysis for option B-A.
Calculate the value of A.
A=$105000−$70000=$35000
Calculate the present worth of benefits.
Substitute $35000 for A, 8% for i and 20 for n in Equation (I).
PWB=$35000[(1+0.08)20−10.08(1+0.08)20]=$350000[(1.08)20−10.08(1.08)20]=$35000(9.8181)=$343634
Calculate the value of C.
C=$800000−$400000=$400000
Calculate the value of L.
L=$100000−$100000=0
Calculate the value of S.
S=$300000−$200000=$100000
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.
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