Expert: Scott A Wilson - 4/27/2011

Question
Ok so, the area of a rectangle is 440 square mm. The height is seven more than three times the base. What is the perimeter??

Answer
Let's give the base length x.
The height is 7 more (which means + 7 to the other side) than 3 times (so there is a 3x).
That gives us the equation of 3x + 7 as the height.

Since the base is of lenth x and the height is of length 3x + 7,
the product gives the area.  So we have x(3x+7) = 440.
Multiply the left side out and get 3x^2 + 7x = 440.
Subtract 440 from both sides and get 3x^2 + 7x - 440 = 0.
This means we need to factors of 440 of which one factor
is 1/3 of the other -7.

The ways that 440 can be split are 1x440, 2x220, 4x110,  5x88, 8x55, 10x44, 11x40, and 20x22.
Multiply the first factor by 3 and add 7.
This gives 1x3 + 7 = 10, not it; 2x3 + 7 = 13, not it;
4x3 + 7 = 19, not it; 5x3 + 7 = 22, not it; 8x3 + 7 = 31, not it;
10x3 + 7 = 37, not it; 11x3 + 7 = 40, and that's it!

This means the pan was 11mm x 40mm.  The perimeter is twice the length
plus twice the width, and that is 2x11 mm + 2x40 mm = 22 mm + 80 mm = 102 mm.