Width of a frame

Question

A rectangular painting​ (not including the​ frame) has a width of

10

inches and a length of

11

inches. If the area of the frame​ (of uniform​ width) is

232

square​ inches, what is the width of the​ frame?

Answer 1

The area of the frame is the are with frame minus the area without the frame.

Then, (10 + 2w)(11 + 2w)- 10 * 11 = 232

110 + 42w + 4w² - 110 = 232

4w^2 + 42w - 232 = 0

2w^2 + 21w - 116 = 0

(2w + 29)(w - 4) = 0

2 = -29/2, 4

We can throw out the negative solution.

Thus, w = 4 in

As a check, 18*19 - 10*11 = 342 - 110 = 232 sq. in.