Derivative and Optimization Masters Come In!?

An architect designs a rectangular garden. The garden is to be surrounded on all sides by a hedge, and a trelissed walkway runs between one pair of opposite sides. The hedge costs $98 per linear foot, and the walkway costs $127 per linear foot. The garden must have a toal of 1000 ft square. What dimensions should the garden have to minimize the cost?

Derivative and Optimization Masters Come In!?
Let l be the length and w be the width of the garden. Then the hedge costs 98(2l+2w) (plus the hedge required to cover the corners, which depends on the width of the hedge, but since that isn't specified, I'll assume the hedge-width is negligable). The walkway, which we shall assume runs across the width of the garden, costs 127w. So the total cost is:



98(2l+2w)+127w

196l + 196w + 127w

196l + 323w



The area of the garden is:

A=lw=1000

So l=1000/w, and the cost of the garden is:



196,000/w + 323w



Taking the derivative and setting it equal to zero:



-196,000/w2+323=0

323w2 - 196,000=0

323w2=196,000

w2=196,000/323

w = √(196,000/323) ≈ 24.633537 feet (we can ignore the negative square root, since that is unphysical).



We note that the second derivative of cost at this point is positive, so this is indeed a minimum. So w=√(196,000/323). Since l=1000/w, l=1000/√(196,000/323) ≈ 40.595064 feet



The total cost of the garden is then:



196l + 323w ≈ $15913.26
Reply:First write a cost function in terms of length and width:



C = 98(2l + 2w) + 127 w



A =lw or l=A/w = 1000/w



C = 196l + 196w + 127 w

C = 196l + 323 w

substituting for l, we get



C = 196000/w + 323w

Take the first derivative

C' = -196000w^-2 +323

Set it equal to 0 and solve

I get approximately 24.6 x 40.6