a rectangular garden 450 square feet in area is to be fenced off against hyenas. find the dimensions that will require the least amount of fencing if one side of the garden is already protected by a barn. a) 30 by 15 feet b) 32 by 14 feet c) 120 by 154 feet d) 29 by 14 feet e) 90 by 5 feet

we trying to minimize L, the length of fencing. Since , one side of the fenced area is covered by the barn, we have got to make two sides perpendicular to the barn plus one side parallel to the barn.

Suppose the length of the perpendicular side be x and the length of the parallel side be y.

Thus, A = x × y = 450 sq. feet --(1)

x + x + y = L --(2)

from (1) , y = 450/x

putting this value in equation (2) we get,

2x + 450/x = L

Now , minimize the L by taking its first derivative with respect to t and then put dL / dt = 0

dL /dx = 2 - 450/x^2

2- 450/x^2 = 0 => 450/x^2 = 2

=> x^2 = 450/2 = 225

taking square root both sides,

=> x = +-√225 = 15 or -15

but sides of any geometry never be negative

so, x = 15 put this x in equation (2)

=> y = 450/15 = 30

Hence , the dimensions of rectangular garden are (15, 30).

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