use the vertex form to find the maximum area the perimeter of a rectangle 400 feet?

Answer 1

The intuitive answer is that a square will have the maximum area. With a perimeter of 400 feet, that would be 100 ft. for each side or 100 * 100 = 10,000 sq. feet for the area.

However, let's prove that using the "vertex form". I think that means to first figure an equation of the area, given the perimeter. That graph will be a parabola. Put the parabola in vertex form and you can read off the answer.

Start with figuring out an equation for the parabola:
Let W = width
Let L = length
You know that 2W + 2L = 400
So W + L = 200

Solving for one of the variables you have:
L = 200 - W

You know the formula for area of a rectangle is:
A = W * L

But since L = 200 - W, you can substitute in L:
A = W * (200 - W)

This can be expanded to give you the formula for your parabola:
A = -W² + 200W


However, we need to get this in vertex form:
Vertex form of a parabola is:
a (x-h)² + k

So first pull out the -1 coefficient on the W² term:
A = -1 (W² - 200W)

For the portion inside the parentheses, you want to complete the square. You do this by taking half of -200 and squaring it (-100)² = 10,000. Add and subtract this from the equation:
A = -1 (W² - 200W + 10000) + 10000

Factor the perfect square and simplify:
A = -1 (W - 100)² + 10000

Looking at the vertex form a (x - h)² + k, you can see your values for h and k are 100 and 10,000 respectively. This means the vertex of the parabola is at (100, 10000). Since it is an upside down parabola, we have indeed found the maximum value.

In other words, when the width of the rectangle is 100 feet, the area will be the maximum of 10,000 sq. ft.

This matches with our intuitive answer of it being a square with sides of 100 ft. each and an area of 10,000 sq. ft.

Answer 2

I have no idea what the "vertex form" might be, but it is a trivial calculus problem to show that the maximal area of a rectangle of given perimeter is a square. In this case, the square would be 100 feet each side, or 10,000 square feet overall.