Word problems

Question

You have 80 yards of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?

Answer 1

80 yards of fencing to enclose rectangular region means the perimeter of this rectangle is 80.

 

P = 80 = 2l+2w

80 = 2(l+w)

l+w=40         eq. 1

 

Area of rectangle = A = l*w      eq 2

 

Solving for l in eq 1,

l = 40-w       eq 3

Substitute eq 3 in eq2 gives us

A = (40-w)(w)

A = 40w - w²

Find critical point of this by finding its derivative and setting it equal to zero gives you the width where are is maximum

dA/dw = 40-2w

Set dA/dw equal to zero to find max,

0 = 40 - 2w

w = 20

Plug w in eq 3,

l = 40-20 = 20

 

Area = l*w = 20*20 = 400

400 yards is the maximum area.