Calc problem?
A landscape architect plans to enclose a 3000 square-foot rectangular region in a botanical garden. She will use shrubs costing $22 per foot along three sides and fencing costing $14 per foot along the fourth side.
Find the minimum total cost.
Round your answer to the nearest integer.
Comments
Let x be the width of the garden in feet (on the fenced side).
Let 3000/x be the length of the garden in feet.
The cost of surrounding the lot will be:
$14 for one width
$22 for two lengths and one width
A function for the cost for a given width is:
f(x) = 14x + 22(x + 2 * 3000/x)
f(x) = 14x + 22x + 84000/x
f(x) = 36x + 84000/x
Take the derivative:
f'(x) = 36 - 84000/x²
Set that to zero:
36 - 84000/x² = 0
Multiply both sides by x²:
36x² = 84000
Divide both sides by 36:
x² = 84000/36
x² = 7000/3
x = ±√(7000/3)
We can ignore the negative value.
x ≈ 60.553
Plug that back in to f(x) and you'll have minimum cost.
Answer:
About $4,360