What is the area of the field within the bleachers?

Question

A rectangular arena has bleachers around the edges. The bleachers are x yards wide. The entire arena (including the bleachers) is 50 yards wide and 100 yards long. What is the area of the field within the bleachers?

What I know:

1. Needs to be in ax^2 + bx + c form
2. area is LxW

How do I set this up to go into quadratic form?

Answer 1

Hi Maximus.

Picture a box within a box. That is what the playing field surrounded on all four sides by bleachers would look like from the air.

A rectangular arena has bleachers around the edges. The bleachers are x yards wide. The entire arena (including the bleachers) is 50 yards wide and 100 yards long. What is the area of the field within the bleachers?

What I know:

Needs to be in ax^2 + bx + c form

area is LxW

You also know:

1. total width = 50 yds
2. total length = 100 yds
3. bleachers are X yds wide

We want to find the area of the field in a quadratic form. 😎 That would be the area of the innner box in our mental picture. OK so far?

To find the area of that box, we need its length and its width:

1. L = 100 - 2X -----> because there are bleachers that are X yds wide in each corner of the rectangle
2. W = 50 - 2X -----> same reason, there are bleachers covering the corners
3. Area = L * W = (100 - 2X) * (50 - 2X)
4. Area = 5000 - 200X - 100X + 4X²
5. Area = 4X² - 300X + 5000 ✔

Looks pretty quad' to me.

Answer 2

The total area is (100)(50)

The dimensions of the filed are (100 - 2x)(50 - 2x) (Draw a diagram!)

A = (100 - 2x)(50 - 2x)

Can you convert to vertex form?