SOLUTION: a picture frame on colored cardboard 40 cm by 32cm. if there is a uniform border around the picture such that area of the border is 720 cm2.find the length and width of the picture

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Question 1186641: a picture frame on colored cardboard 40 cm by 32cm. if there is a uniform border around the picture such that area of the border is 720 cm2.find the length and width of the picture.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The cardboard is 8cm longer than it is wide. Since the border is uniform, the picture itself is 8cm longer than it is wide.

The area of the cardboard is 40*32=1280 cm^2; the area of the border is 720 cm^2. So the area of the picture is 1280-720=560 cm^2.

Let x be the width of the picture; then x+8 is the length. Since the area is 560...

x%28x%2B8%29=560
x%5E2%2B8x=560
x%5E2%2B8x-560=0
%28x%2B28%29%28x-20%29=0

x=-28 or x=20

Obviously the negative answer makes no sense in the problem. So x=20.

ANSWER:
picture width: x=20cm
picture length: x+8=28cm

Note in solving the problem using formal algebra we had to factor a quadratic expression by finding two numbers whose difference is 8 and whose product is 560. That's what the original problem required us to do; so the formal algebra didn't get us any closer to the answer.

Trial and error and a bit of mental arithmetic will get us to the solution quickly:
560=56*10 no...
560=14*40 no...
560=28*20 YES!