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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?

2006-10-16 15:50:43 · 2 answers · asked by ? 1 in Science & Mathematics Mathematics

2 answers

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

2006-10-16 16:12:03 · answer #1 · answered by Pascal 7 · 0 0

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

2006-10-16 16:28:04 · answer #2 · answered by Uncle Bill 2 · 0 0

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