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The frame around a rectangular picture has a uniform width. The dimensions of the picture are 5 in. by 7 in. If the combined area of the picture and the frame is 80 in.2, what is the width of the frame?

 

 Mar 25, 2021
 #1
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First, find the area of the picture:

5*7 = 35 in2

 

The area of just the frame is:

80 - 35 = 45 in2

 

Now, note that the frame is nothing but 4 equal squares for the corners of the frame and 4 rectangles.

 

The corners have an area of x2.

Two of the rectangles have an area of 5x, and the other 2 have an area of 7x.

Therefore the area of the frame is

4(x2) + 2(5x) + 2(7x) = 4x2 + 24x

Which equals:

4x2 + 24x = 45

 

Solving the quadratic, we get:

4x2 + 24x = 45

4x2 + 24x - 45 = 0

(2x-3)(2x+15) = 0

 2x - 3 = 0

2x = 3

x = 1.5

(The other solution for x would be negative)

 

Therefore, the width is 1.5 in2.

Solved! :)

 Mar 25, 2021
 #2
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(2x + 5)(2x + 7) = 80

 

x = 1.5

 Mar 25, 2021

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