Question

you have a photograph 12 in. wide and 18 in. long you surround the photograph with mat x in. wide you want to make a frame for the matted photo but you only have an 80 inch length of wood to use for the frame what are the dimensions for the largest frame you can make

Answer 1

The dimensions of the original picture are 12 by 18. Adding the mat of width x around the picture makes the new dimensions of the whole thing (18+2x)(12+2x). Now we need to find x, the width of the mat, in order to then find the new dimensions. We want the largest frame, so we will assume all 80 inches of wood are used.
To find x, notice that the total area - the original picture area = the area of just the wood used. This must equal 80. So let's set up an equation and solve for x.

(18 + 2x) (12 + 2x) - 18 * 12 = 80
216 + 36x + 24x + 4x^2 - 216 = 80
4x^2 + 60x = 80
4x^2 + 60x - 80 = 0
Divide by GCF, which is 4, to simplify the equation:
x^2 + 15x - 20 = 0
Now this can't be factored, so you must either use the quadratic formula or plug it into your graphing calculator and find the x-intercepts.
EIther way the zeros are x = 1.2321, and x = -16.2337
But this is a word problem so we can't have a negative length for x. So the only valid solution for x is 1.2321. This is the width of the mat all around the 12 * 18 picture.

Now let's use x to find the new dimensions:
Remember that the new dimensions are formed by adding 2x (that is, 2 times x, with x being the width of the mat) to the length and width of the original picture.
New width with mat = 12 + 2x = 12 + 2*1.2321 = 14.46 in
New length with mat = 18 + 2x = 18 + 2.1.2321 = 20.46 in

Answer 2

Here's a diagram to help: