Question

Lesson 18.2- Multiplying Polynomial ExpressionsLesson 18.3 - Special Products of BinomialsTHE TWO PURPLE PROBLEMS Kyra is framing a square painting with side lengths of (x + 8) inches. The total area of thepainting and the frame has a side length of (2x - 6) inches. The material for the frame will cost$0.08 per square inch. Write an expression for the area of the frame. Then find the cost of thematerial for the frame if x = 16.

Answer 1

Given that:

- Kyra is framing a square painting with side lengths (in inches) of:

`(x+8)`

- The total area of the painting and the frame has a side length (in inches) of:

`(2x-6)`

• You need to remember the formula for calculating the area of a square is:

`A_(square)=s^2`

Where "s" is the side length of the square.

Therefore, you can determine that the area of the square painting (in square inches) is:

`A_p=(x+8)^2`

And the total area (in square inches) of the square frame and the square painting is:

`A_t=(2x-6)^2`

Therefore, the area of the frame is the Difference between them:

`A_f=(2x-6)^2-(x+8)^2`

You can simplify the area as follows:

1. Apply this formula:

`(a\pm b)^2=a^2-2ab+b^2`

Then:

`A_f=(2x)^2-(2)(2x)(6)+(6)^2-\lbrack(x)^2+(2)(x)(8)+(8)^2\rbrack`
`A_f=4x^2-24x+36-x^2-16x-64`

2. Add the like terms:

`A_f=3x^2-40x-28`

• You know that the material for the frame will cost $0.08 per square inch. Then, in order to find the cost of the material is:

`x=16`

You first need to substitute that value into the equation for the area of the frame and evaluate:

`A_f=3(16)^2-40(16)-28=100`

Multiply the cost $0.08 by that area:

`(\text{ \$}0.08)(100)=\text{ \$}8`

Hence, the answer is:

- Expression for the area of the frame:

`3x^2-40x-28`

- Cost of the material for the frame:

`\text{ \$}8.00`