A rancher is fencing off a rectangular area with a fixed perimeter of 76m. What dimensions would yield the maximum area? What is the maximum area?

Question

So could you answer this by completing the square, Thank you so much for your help!

Answer 1

Assume that the length = x, and the width= y

2x + 2y = 76

y = 76/2 - x

y = 38 - x

Area = x y

= x ( 38 - x)

= 38 x - x²

= - ( x² - 38 x )

= - ( x² - 38 x + 19² ) + 19²

= - ( x - 19 ) ² + 361

This equation is an equation for a downward parabola ( because we have minus sign in front of term x²) with vertex ( h, k),

h=19 which represent the shifting to the right in the x direction, and k=361 represent the shifting upward in the y- axis , which means that we have maximum area occur when x= 19 and it is equal 361m²

There is another way

Area = 38 x - x²

drive A for x

dA/dx = 38-2x

at max dA/dx = 0

38- 2x=0

x =19 , y =38- x= 38 -19 =19

max A = 19*19 = 361

Answer 2

length + width = 38

length = 38 - width

w represents width

38 - w represents length

A = w(38 - w)

A = -w² + 38w

A = -(w² + 38w + 19²) + 19²

A = -(w + 19)² + 361