Question

A rancher wants to fence an area of 1,500 square yards in a rectangular field that borders a straight river, and then divide it in half with a fence perpendicular to the river. he needs no fence along the river. the dividing fence costs half as much as the surrounding fence. how can he do this so that the cost of the fence is minimized? follow the steps:

Answer 1

Let the width of the area be x and  length be y. If the river runs along y and cost of fencing the surrounding lengths is unit,

Cost, C=2x+y+1/2x = 2.5x+y
Area, A= 1500= xy ---> y=1500/x
C= 2.5x+1500/x
At minimum cost, the first derivative of C function is equal to 0
That is,
dC/dx =0 = 2.5-1500/x^2 => x =sqrt (1500/2.5) = 24.49 yard
Then, y=1500/x = 1500/24.49 = 61.24 yard
Therefore, for lowest cost of fencing, the width should be 24.49 yard and length be 61.24 yards.