Part A: Which equation represents the combined area of the photo and the mat? Part B: What is the factored form of the equation that represents the combined area of the photo and the mat?

Question

Leonard needs to design a mat with an  opening for a photo that he is going to frame. He wants the mat to have the same width on all four sides, as shown in the following figure.

© 2017 FlipSwitch. Created using GeoGebra.

The combined area of the photo and the mat needs to be  square inches.

Part A: Which equation represents the combined area of the photo and the mat?

Part B: What is the factored form of the equation that represents the combined area of the photo and the mat?

Answer 1

The total area of Leonard's photo and mat is the total area of the large rectangle that the photo and mat make.

Remembering, the area of a rectangle = length x width, we can substitute what we know so far and form an equation.

The length will be the 8" height of the photo plus the width of the mat on the top and the width of the mat on the bottom (2 widths). Thus the length becomes 8+2w.

Likewise, the width will be the 10" width of the photo plus the width of the mat on the right and the width of the mat on the left (2widths). Thus the width becomes 10+2w.

So, length x width = (8+2w)(10+2w).

Multiplying the 2 binomials, we get (8x10) + (8x2w) + (2wx10) + (2w x 2w).

Now we have 80w + 16w + 20w + 4w².

Combining like terms and rearranging our terms, our equation becomes

4w² + 36w + 80 = A (total area) If it is a multiple choice question, which it sounds like it is, this is what your equation will look like. OR it will look like a 4 has been factored out and then it would be 4(w²+9w+20) = A. This is explained in the next paragraph.

To factor our equation, we first will factor out any common factors. Our only common factor in this equation is a 4, because each term is divisible by 4.

After we divide each coefficient by 4, our equation looks like this.

4(w²+ 9w +20) =A

Now we can factor our trinomial to finish completely factoring the equation.

We are looking for 2 numbers that we can multiply to get 20 and add (because we have 2 plus signs in our equation) to get 9. They are 4 and 5.

Now our equation becomes 4(w+4)(w+5) = A which is completely factored.

Hope this helps :)

Answer 2

The photo and the mat make up,the entire figure

The dimensions of the figure are (8+2u)(10+2u) so your equation is

Area = (8+2u)(10+2u)

Answer 3

photo=8*10=80sq/in

frame=(8+2μ)(10+2μ)