The length of a picture without its border is....

Question

The length of a picture without its border is 6 inches less than twice its width. If the border is 1 inch wide, and its area is 70 square inches, what are the dimensions of the picture alone?

Answer 1

L = length of picture

W = width of picture

L = 2W-6

Border is 1" all around so

 

Length of border is L + 2 (add one inch to each end)

Width of border is W + 2 (add one inch to each end)

 

Area of border is area of picture with border less area of picture without border

 

Area of picture with border = (L+2)(W+2)

Since L = 2W -6, we can rewrite this expression as

(2W-6+2)(W+2) = (2W-4)(W+2) = 2W² - 4W + 4W - 8 = 2W²-8

 

Now the area of the picture alone (without the border) is

(L)(W) 

Using the same substitution, we have

(2W-6)(W) = 2W²-6W

 

The difference between these two bolded expressions is the area of the border.

2W² - 8 - (2W²-6W) = 70

 

Which is equivalent to

6W - 8 = 70

 

We can solve this linear equation for W

6W = 78

W = 13 inches

 

Finally we can determine L since L = 2W - 6

L = 2(13) -7

L = 26-6 = 20 inches

 

So the picture alone is 20" x 13"