Find the dimensions and the minimum perimeter

Question

Find the dimensions of the rectangle with area 196 square inches that has minimum perimeter, and then find the minimum perimeter.

Answer 1

Since the area is 196 in², LW = 196 => L=196/W or 196W^-1.

P=2L+2W

P=2(196)W^-1 + 2W

dP/dW = -392W^-2+2

That derivative equals 0 when:

392/W²=2

2W²=392

W²=196

W=±√196=±14, rejecting -14.

So, L=14 too (a square).

The perimeter is: P = 2(14)+2(14)=56 in.

The 2nd derivative test could be used to show that indeed W=14 produces a minimum perimeter.

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