SOLUTION: A rectangular painting is to have a total area (including the frame) of 1200 cm2. If the painting is 30 cm long and 20 cm wide, find the width of the frame.

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Question 1194518: A rectangular painting is to have a total area (including the frame)
of 1200 cm2. If the painting is 30 cm long and 20 cm wide, find
the width of the frame.
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

x = width of the frame in centimeters.
This is some positive real number, meaning that x > 0

The width of 30 cm bumps up to 30+x+x = 30+2x cm
This is because we add on two copies of x for either side along the 30 cm dimension.
The same thing happens with the 20 cm bumping up to 20+2x cm.

The painting + frame has dimensions of (30+2x) cm by (20+2x) cm

Multiply those dimensions and set the product equal to the stated total area of 1200 sq cm.

length*width = area
(30+2x)*(20+2x) = 1200
600+60x+40x+4x^2 = 1200
600+100x+4x^2 = 1200
4x^2+100x+600-1200 = 0
4x^2+100x-600 = 0
4(x^2+25x-150) = 0
x^2+25x-150 = 0/4
x^2+25x-150 = 0

Now use the quadratic formula to find x.
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%2825%29%2B-sqrt%28%2825%29%5E2-4%281%29%28-150%29%29%29%2F%282%281%29%29

x+=+%28-25%2B-sqrt%281225%29%29%2F%282%29

x+=+%28-25%2B-++35%29%2F%282%29

x+=+%28-25%2B35%29%2F%282%29 or x+=+%28-25-35%29%2F%282%29

x+=+%2810%29%2F%282%29 or x+=+%28-60%29%2F%282%29

x+=+5 or x+=+-30
Ignore the negative solution. Recall at the top we stated that x > 0.

Answer: The width of the frame is 5 cm.