Answer the following:

Question

The perimeter of a rectangle is 72 meter, find its dimension and its maximum area

Answer 1

The perimeter of a rectangle is 72 meter, find its dimension and its maximum area

Detailed Solution:

Given/Known: Perimeter of a rectangle = 2x + 2y = 72, Area = A = xy

2x + 2y = 72 ==> x + y = 36 ==> y = 36 – x

A = xy = x(36 – x) = 36x – x² ==> A = 36x – x²

Area of rectangle in vertex format to see the maximum at vertex: (h, k) = (x, A)

A = a(x – h)² + k

= 36x – x²

= –x² + 36x

= –(x² – 36x)

= –(x² – 36x + 324 – 324)<==completing square:(b/2)² = (-36/2)² = (-18)² = 324 (add & subtract this)

= –((x – 18)² – 324)

= –(x – 18)² + 324

Note: We arrive at area of rectangle in vertex format that opens down (concave down) means the vertex is maximum: (h, k) = (18, 324) = (x, A)

Dimensions that give maximum area are: (x, y) = (18, 18)

x = 18

y = 36 – x ==> y = 36 – 18 ==> y = 18

Maximum Area: A = xy ==> A = (18)(18) ==> A = 324 <== Final Answer

Answer 2

The simplest approach to this problem - to use the derivative, although intuition is telling us that we will have a square anyway.

Let x be the width of the rectangle.. Knowing that perimeter is 72 m, and it is the sum of 2 width and 2 length, we concluded that the length of the rectangle is 36 - x, m.

The area is width times length, or A = x(36 - x) = 36x - x²

Let's find derivative of this expression with respect to x.

dA/dx = 36 - 2x.

We will have an extremum of this expression, when it is equal to zero. So 36 - 2x = 0.

it is easy to see that x = 18 m. That was expected. We got the result: The maximum area, when the rectangle is a square with the side of 18 m. And it is 324 m².

You can use the second derivative test, to make sure that the function has maximum!

Hope that my explanation was helpful.