Problema Solution
Among all rectangles that have a perimeter of
144 , find the dimensions of the one whose area is largest. Write your answers as fractions reduced to lowest terms
Answer provided by our tutors
Let
x = the width
y = the length
2(x + y) = 144 divide both sides by 2
x + y = 72
y = 72 - x
The area of the rectangle:
A = x*y
Plug y = 72 - x into A = x*y:
A = x*(72 - x)
A = -x^2 + 72x
We need to find such x so that A has maximum:
x max = -b/2a, where a = -1, b = 72
x max = -72/(2*(-1))
x max = 36
y = 72 - 36
y = 36
The dimensions of the rectangle with the largest area are:the length and the width are equal to 36 (the rectangle is square).