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Question
The area of a rectangle is 60 square meters. If each side is increased by 3 meters,the area of the square on the diagonal is increased by 120 square meters .find the dimensions of the original rectangle?
If there is a rectangle, let's say that it is x by y.
The area of the rectangle is 60, so xy = 60.
If each side is increased by 3 meters,
the dimension is now (x+3)(y+3).
The area of the square of the diagonal is the result of the Pythagoreus Theorem. It is the diagonal squared.
The diagonal is the √((x+3)² + (y+3)²), but this is squared,
so the area is (x+3)² + (y+3)².
It says that the area of the diagonal is increased by 120,
so x² + y² + 120 = (x+3)² + (y+3)².
Squaring out the right side gives
x² + y² + 120 = x² + 6x + 9 + y² + 6y + 9.
The x² + y² can be subtracted from both sides,
so we have 0 = 6x + 6y - 102.
Now since xy = 60, y = 60/x. Substitue this in and the result is
0 = 6x + 360/x - 102. Multiply everything by x/6 and you get
0 = x² + 60 - 17x, which is the same as 0 = x² - 17x + 60.
This factors into (x-5)(x-12).
If x is 5, y is 12. If x is 12, y is 5.
Either way, the size is 60 and the dimensions are 5 x 12.
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Scott A Wilson
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