Practice question for a test. I have no idea how to do this, but I do have the answer

Question

Hello!

I am practicing for an upcoming test I have to take, and I have a practice test with practice problems and answers.

The question I'm having trouble with is "A rectangular poster is three times as high as it is wide. If it contains a picture of 125 square inches framed by a margin of 4 inches at the top and bottom and 3 inches on each side, find the width of the poster."

The answer to this problem is apparently 11, however I cannot figure out how one gets this answer. Thank you!

Answer 1

Let the width of the poster be w. The height is three times the width, so the height is 3w. The picture is surrounded by a margin of 4 in. at the top and 4 in. at the bottom, so the height of the picture is 3w - 4 - 4 = 3w - 8. Similarly, the picture is surrounded by a margin of 3in. on the left and 3in. on the right, so the width of the picture is w - 3 - 3 = w - 6.

Now, the picture has a height of 3w - 8 and a width of w - 6 so the area of the picture is

A = (3w - 8)*(w - 6) = 3w² - 26w + 48

Since we were given that the area of the picture is 125 in.², we can write

3w² - 26w + 48 = 125

3w² - 26w - 77 = 0

Using the quadratic formula,

w = (26 ± √((26)² - 4*3*(-77)))/2*3

w = (26 ± √1600)/6

w = (26 ± 40)/6

The positive solution will yield

w = 66/6 = 11 in.

Answer 2

Let x = width and 3x = height

Dimensions of picture are x - 6 and 3x - 8

(x - 6)(3x - 8) = 125

3x² - 26x - 77 = 0

(x - 11)(3x + 7) = 0

x = 11 or x = -7/3 Since x can't be negative, x must equal 11.