Question

Would appreciate if someone checked my optimization answer for:
John, the manager of a store, wishes to add a fenced-in rectangular storage yard of 20,000 square feet, using the building as one side of the yard. Find the minimum amount of fencing that must be used to enclose the remaining 3 sides of the yard.

My answer: 400 yards

Answer 1

let x and y be the sides of rectangle.
xy=20,000
y=20000/x
\[L=2x+y,=2x+\frac{ 20,000 }{ x }\]
where L is the length of fence.
\[\frac{ dL }{ dx }=2-\frac{ 20000 }{ x^2 }\]
\[\frac{ dL }{ dx }=0,gives\]
\[2-\frac{ 20000 }{ x^2 }=0,gives \]
\[2x^2=20000,x^2=10000,x=100\]
\[\frac{ d^2L }{ dx^2 }=-\frac{ -2 \times 20,000 }{ x^3 }=\frac{ 40,000 }{ x^3 }\]
at x=100
\[\frac{ d^2L }{ dx^2 }=\frac{ 40,000 }{ 100^3 }>0 \]
Hence L is minimum when x=100
\[y=\frac{ 20000 }{ 100 }=200\]
Hence minimum length of fence=2x+y=2*100+200=400 yards
so you are correct.

Answer 2

Thank you.

Answer 3

yw