A rectangular picture is surrounded by a border which has uniform width 2 inches. Without the border the length of the picture is twice its width.
If the area of the border is 196 square inches, what is the length, in inches, of the picture, excluding the border?
Answer: 30
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mschling52
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width of picture = x
length of picture = 2x
width of picture+border = x+4 (2 inches on each end)
length of picture+border = 2x+4 (2 inches on each end)
If you draw this out it will be easier to visualize, but the area of the border alone is given as 196. This is also equal to the area of the picture+border minus the area of the picture alone. We can solve for x by setting 196 equal to a formula for the area of the picture+border minus the area of the picture alone...
196 = (2x+4)(x+4) - 2x^2
196 = 2x^2 + 8x + 4x + 16 - 2x^2
196 = 12x + 16
180 = 12x
15 = x
Then, since we are asked to find the length of the picture alone, which we said above was equal to 2x, we get 2x = 2(15) = 30
length of picture = 2x
width of picture+border = x+4 (2 inches on each end)
length of picture+border = 2x+4 (2 inches on each end)
If you draw this out it will be easier to visualize, but the area of the border alone is given as 196. This is also equal to the area of the picture+border minus the area of the picture alone. We can solve for x by setting 196 equal to a formula for the area of the picture+border minus the area of the picture alone...
196 = (2x+4)(x+4) - 2x^2
196 = 2x^2 + 8x + 4x + 16 - 2x^2
196 = 12x + 16
180 = 12x
15 = x
Then, since we are asked to find the length of the picture alone, which we said above was equal to 2x, we get 2x = 2(15) = 30
