A landscape engineer has 200 feet of border to enclose a rec

A landscape engineer has 200 feet of border to enclose a rectangular pond. What dimensions will result in the largest pond? A special window in the shape of a rectangle with semicircles at each end is to be constructed so that the outside dimensions are 100 feet in length. See the illustration. Find the dimensions of the rectangle that maximizes its area.

Solution

2)

let the length of pool be x , width of pool be y

perimeter of rectangle= length of border

2(x+y)=200

x+y =100

x =100-y

for largest pond area is maximum.

area of pond =length *width

A=x*y

A=(100-y)*y

A=100y-y²

form maximum area dA/dy =0, d²A/dy²<0

=>100 -2y=0

=>y =50

d²A/dy² =-2

x =100 -y , y =50

=>x =100-50

=>x =50

dimensions of largest pond are length =50 ft , width =50ft