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
[tex]\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[/tex]
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 [tex]n=log_2(16)=4[/tex]
We can write the cost of the 16th house as
[tex]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[/tex]
The estimated material cost per square foot for the last (16th) home is $19.34.
