A farmer wants to construct a fence around a rectangular field. Sides with neighbours need reinforced fencing that costs $8 per meter. The other sides use regular fencing that costs $4 per meter. Assuming the farmer has neighbours on the east and west sides, answer the following questions. Give your answers as exact expressions or decimals accurate to at least two places.

Question

a)  Given a budget of $500, find the dimensions that give the maximum area inside the fencing.

Length of east and west fences:

b)  Find the dimensions that will cost the least to build a fence with an area of 1878 m2.
Length of east and west fences:

Thanks

Steve · Answer

if the NS sides have length x and the EW sides have length y, then if he spends the entire $500,

2*4x + 2*8y = 500
or,
2x+4y=125

A = xy = x(125-2x)/4

This is just a parabola, with its vertex (maximum area) at (125/4, 15625/32)

For #b, y=1878/x, so the cost c is

8x+16y = 8x+16(1878/x)

This has a minimum at (2√939, 32√939)