Question

40 points!

You are framing a picture with a frame of equal width on each side.

a. Write a polynomial that represents the area of the picture including the frame

b. Find the area of the picture including the frame when the width of the frame is 2 inches

Answer 1

Part A)

The polynomial is:

`A=4(x+8)(x+10)`

Or, expanded:

`A=4x^2+72x+320`

Part B)

When the width of the frame is two inches, the area of the picture including the frame is 480 square inches.

Explanation:

Part A)

For solely the picture, the width is 16 inches, and the length is 20 inches.

If we include the width of the frame, then we will have to add x to both sides of the 16 inches. Then the new width will be (16 + 2x) inches.

Likewise, the new length will be (20 + 2x) inches.

The area of a rectangle is given by its width multiplied by its length. Then the polynomial that represents the area will be:

`A=(16+2x)(20+2x)`

To simplify calculations, we can factor out a two from both factors. Thus:

`A=2(8+x)\cdot 2(10+x)=4(x+8)(x+10)`

Expanded, this will be:

`A=4(x^2+18x+80)=4x^2+72x+320`

Part B)

When the width of the frame is two, x = 2. Then the area is:

`A=4(2+8)(2+10)=4(10)(12)=40(12)=480\text{ inches}^2`

The area of the picture including the frame will be 480 square inches.