Question

The Betterbilt Construction Company designs and builds residential mobile homes. The company is ready to construct, in sequence, 16 new homes of 2,400 square feet each. The successful bid for the construction materials in the first home is $64,800, or $27 per square foot. The purchasing manager believes that several actions can be taken to reduce material costs by 8% each time the number of homes constructed doubles. Based on this information, a. What is the estimated cumulative average material cost per square foot for the first five homes? b. What is the estimated material cost per square foot for the last (16th) home?

Answer 1

Answer:

(a) The estimated cumulative average material cost per square foot for the first five homes is $24.47.

(b) The estimated material cost per square foot for the last (16th) home is $19.34.

Explanation:

(a) If the cost its reduced by 8% every time the number of homes is doubled, we can express the cost of the first five houses as

C1 = C

C2 = C*(1-0.08)=0.92*C

C3 = C2 = 0.92*C

C4 = C2*(1-0.08)=0.92*0.92*C = 0.8464*C

C5 = C4 = 0.8464*C

Then, the average cost of the first five houses is

$\bar{C}=(1/5)*(C1+C2+C3+C4+C5)\\\\\bar{C}=(1/5)*(C+0.92C+0.92C+0.8464C+0.8464C)\\\\\bar{C}=(1/5)*4.5328*C = 0.90656*C=0.90656*27=24.47$

The estimated cumulative average material cost per square foot for the first five homes is $24.47.

For the 16th home, the number we can estimate that the number of homes double 4 times: at house number 2,4, 8 and 16.

Other way to calculate that is $n=log_2(16)=4$

We can write the cost of the 16th house as

$C_{16}=0.92*C_8=0.92^{2} *C_4=0.92^{3} *C_2=0.92^{4} *C\\\\C_{16}=0.92^{4} *C=0.716*C=0.716*27=19.34$

The estimated material cost per square foot for the last (16th) home is $19.34.