Area List

question continued….and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence? (Give the dimensions in increasing order.)

Let one side be length A and the other B
Total area = A.B = k where k is the constant value 37.5 million

Then B = k/A……………….(i)

The total length of the sides plus the division fencing will be

L = 2A + 2B + A if we let the division fence be parallel to A: it doesn’t matter either way.

So L = 2A + 2k/A + A = 3A + 2k/A …………….. (ii)

Differentiate L with respect to A, to get

L’ = 3 – 2k/(A^2) = 0 for max or min

This gives A^2 = 2k/3 and so A = sqr root of 2k/3

Substitute in (i) to get

B = k/[sqr root of 2k/3] = sqr root of 3k/2 by division and rearrangement.

Now put the value of k in place as 37.5 million and
the length of side A should be 5000 ft and
the length of side B should be 7500 ft and the division fence should be parallel to side A and measure 5000 ft.

Check the second derivative of L, to get

L” = 4k/(A^3) which is positive since k and A are both positive, so we have a minimum.

The total length of wire will be 30,000 feet by substituting values in (ii) and this is the minimum quantity and therefore minimum cost.

OK?