Calculus Optimization Problem!?
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A landscape architect plans to enclose a 3000 square foot rectangular region in a botanical garden. She will use shrubs costing $25/foot along three sides and fencing costing $10/foot along the 4th side. find the minimum total cost.
Please help and explain with steps. Thanks!
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LW = 3000
25(2L) + 25W + 10W —> 50L + 35W —-> minimum
Solving the first equation for L, and substituting into the 2nd equation, we have:
Cost = C(W) = 50(3000/W) + 35W = 150000/W + 35W
C’(W) = -150000/W^2 + 35
C’ = 0 when W=65.46536707
C(65.465) = 4582.58
The minimum total cost is $4582.58. This occurs when the width is 65.465 feet and the length is 45.826 feet.
Find a cost function then an area function…
Input the cost function into the area function so that you have one equation in total
Find the derivative
Set the problem to 0 and solve for x or y
with your x or y value then plug it into the original function and solve for the other variable