Question

1. You are looking to spruce up your garden by making a rectangular enclosure using a wall as one side and 160 meters of fencing for the other three sides. You want to find the dimensions of the rectangle so that you are maximizing the enclosed area.

(a) Draw and label a picture representing the problem

(b) Write the objective function and the constraint. (You do not need to label which equation is which.)

(c) Write the area equation in terms of one variable.

(d) Find the dimensions that give the greatest area.

(e) What is the greatest area that can be enclosed?

Answer 1

Let x,y be legnth of sides

As one side is along to wall and other three sides are fenced with 160 m of fence

2x+y=160 (constraint )

y=160 - 2x ----------------(1)

Area of rectangle

A=xy -------(2) (Objective function)

=x(160-2x) (putting the value of y from equation (1))

=160x-2x^2 ---------(3) (Objective function in one variable)

Differentiating with respect to x

dA/dx = 160 -2*2x

=160 - 4x

For critical points

dA/dx=0

160 - 4x=0

4x=160

x=40 meter

d^2A/dx^2 = -4<0 so x=40 is maxima point

from the equation (1)

y=160-2*40=80 meter

So dimension are x=40 m and y=80 m

Amax = 40*80=3200 m^2