Question 116771: A rectangle having area 120 square meters has sides of length X+3 meters and x+10 meters. What is the length of a diagonal of the rectangle?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! The area of a triangle is equal to its length times its width. For this problem the width
is (x+3) meters and the length is (x + 10) meters. The product of these two is the area and, therefore,
is equal to 120 square meters. So we can write:
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(x + 3)(x + 10) = 120
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Multiplying out the left side results in:
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x^2 + 13x + 30 = 120
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To solve this equation, get it into the standard form of:
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ax^2 + bx + c = 0
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by subtracting 120 from both sides to get rid of the 120 on the right side. When you do this
subtraction to both sides you get:
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x^2 + 13x - 90 = 0
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Factor the left side to get:
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(x + 18)(x - 5) = 0
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This equation will be true if either of the two factors is equal to zero because a multiplication
by zero on the left side will make the left side equal the zero on the right side.
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Set the two factors equal to zero (one at a time) and solve for x. First:
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x + 18 = 0 ... subtract 18 from both sides and get that
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x = -18
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That won't work because if x = -18, then each of the two sides would have a negative
size. Negative width and negative length doesn't make sense.
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Next set the second factor equal to zero:
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x - 5 = 0
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solve for x by adding 5 to both sides to get rid of the -5 on the left side. This makes
the equation become:
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x = 5
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That looks good. When x = 5, then the width (which is x + 3) is 8. And the length (which
is x + 10 = 15.
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So the sides of the rectangle are 8 and 15. When you draw the diagonal of the rectangle, then
it forms the hypotenuse of a right triangle which has legs of 8 and 15. By the Pythagorean theorem
you know that the sum of the squares of these two legs equals the square of the hypotenuse or
diagonal. In equation form this is:
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8^2 + 15^2 = H^2
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Squaring out the terms on the left side results in:
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64 + 225 = H^2
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Add the terms on the left side and get:
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289 = H^2
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Solve for H by taking the square root of both sides to get:
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17 = H
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Since all the dimensions in this problem are in meters (or meters^2) the answer is that
the length of the Diagonal of the square is 17 meters.
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Hope this helps you to understand the problem and how to go about getting the answer to it.
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