Answer:
A. $ 6053.44
B. From the graph, R = 21.027 ft and C = $ 5700.005, L = 31.043 ft
The values in A and B do not all agree. This could be due to error in approximations. Their values are close though.
Step-by-step explanation:
A. The area A of the enclosure equals, A = 2RL + πR²/2.
The total cost C = 20 × length of curved section + 30 × length of straight section
C = 20πR + 30[2(L + 2R)]
= 20πR + 60L + 120R
Making L subject of the formula from A,
L = A/2R - πR/4
Substituting L into C, we have
C = 20πR + 60(A/2R - πR/4) + 120R
= 20πR + 30A/R - 15πR + 120R
= 5πR + 30A/R + 120R
We now differentiate C with respect to R to find the value of R for minimum cost
dC/dR = 5π - 60A/R² +120
Equating dC/dR to zero, we have
5π - 60A/R² + 120 = 0
So, R = ±√[60A/(5π + 120)]
substituting A = 2000 ft²
R = ±√[60 × 2000/(5π + 120)] = ±29.74 ft
We take the positive answer, R = 29.74 ft since R cannot be negative.
To determine if this is a minimum point, we differentiate dC/dR with respect to R.
So d²C/dR² = 120A/R³
Since d²C/dR² = 120A/R³ > 0 for positive R, it is a minimum point.
Substituting the value of R into C we have
C = 5πR + 30A/R + 120R
= 5π(29.74) + 30 × 2000/29.74 + 120(29.74)
= 467.155 + 2017.485 + 3568.8
= $ 6053.44
and L = A/2R - πR/4
= 2000/2(29.74) - π(29.74)/4
= 33.625 - 23.356
= 10.269
≅ 10.27 ft
B. From the graph, R = 21.027 ft and C = $ 5700.005, L = 31.043 ft
The values in A and B do not all agree. This could be due to error in approximations. Their values are close though.
