Final answer:
The thickness of the picture frame is found by solving the equation derived from the outer dimensions of the frame and the area of the visible picture. The result is a frame thickness of 2.5 inches.
Step-by-step explanation:
To find the thickness of the frame, we first need to understand that the area of the frame and the visible picture together equal the product of the outside dimensions given (13 in. by 14 in.). We then subtract the area of the picture from the area of the larger rectangle (the outer dimensions of the frame) to find the area of the frame alone.
Let's denote the thickness of the frame as x inches. The dimensions of the visible part of the picture are the outer dimensions minus twice the thickness of the frame because there is a frame on both sides. Thus, the dimensions of the picture are (13 - 2x) by (14 - 2x). Given that the area of the visible picture is 132 in², we can establish the following equation:
(13 - 2x)(14 - 2x) = 132
Solving for x:
182 - 26x - 28x + 4x² = 132
4x² - 54x + 182 - 132 = 0
4x² - 54x + 50 = 0
We then use the quadratic formula or factoring to solve for x, but note that since we are looking for a physical dimension, we will only take the positive root as the answer.
If we assume the quadratic factorizes easily, we get:
(2x - 5)(2x - 10) = 0
So the possible values for x are 2.5 and 5. Since twice the thickness cannot be greater than the smallest side of the frame (13 inches), the only logical answer is x = 2.5 inches.
Therefore, the thickness of the frame is 2.5 inches.