A fenced enclosure consists of a
rectangle (length L and width 2R)
attached to a semicircle with a radius R
as pictured below. Note: there is
fencing between the rectangular and
semi-circular portions. The enclosure
is to be built to have a total area (the
entire shaded region), A, of 2000 ft2.
The cost of the fence is $20/ft for
curved sections and $30/ft for straight
sections. Analytically (show all of your
steps using algebra and calculus in the
write up), find the minimum cost to
build the fence and the dimensions of
the enclosure. Show all equations that
you derive and use. This part should
be done entirely in the write-up: no
coding. It is fine to do this by hand and
include a high-resolution picture in the
writeup rather than using Equation
Answers
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.
Hope this is helpful
