Question

Jessie is building a cabinet door, which is pictured below. The center of the cabinet door is 12 inches by 16 inches, and the border of the cabinet door has a width of x inches. The entire cabinet door has an area of exactly 396 square inches.

Clockwise from the top, the picture reads, x, 12 inches, 16 inches, x, Figure not drawn to scale.


Write an equation in standard form that can be used to find x , the width of the border.

Answer 1

`x^2+14x-51=0`

Explanation:

Since the entire cabinet door is a rectangle, its area is given by:

`A=w\ell`

The center of the cabinet door measures 12 inches by 16 inches.

And the border of the cabinet door has a width of x inches.

Therefore, the total width of the cabinet door will be (12 + 2x).

Likewise, the total length of the cabinet door will be (16 + 2x).

Thus, the area of the entire cabinet door will be:

`A=(2x+12)(2x+16)`

We are given that this entire area is 396 square inches. Thus:

`396=(2x+12)(2x+16)`

Simplify. We can factor out a two from both of the factors on the right-hand side:

`396=2(x+6)(2)(x+8)`

So:

`396=4(x+6)(x+8)`

Dividing both sides by four yields:

`99=(x+6)(x+8)`

Distribute the right-hand side:

`x^2+14x+48=99`

Therefore, our equation in standard form is:

`x^2+14x-51=0`

Edit:

We want to find the width of the board. So, we simply have to solve for x. We previously obtained:

`x^2+14x-51=0`

We can factor:

`(x+17)(x-3)=0`

By the Zero Product Property:

`x+17=0\text{ or } x-3=0`

Solve for both cases:

`x=-17\text{ or } x=3`

Length/width cannot be negative. So, we can reject the first solution.

Thus, our only solution is:

`x=3`

The width of the border of the cabinet door is 3 inches.

This means that the total length of the cabinet is 16 + 2(3) = 22 inches, and the total width of the cabinet is 12 + 2(3) = 18 inches.