Question 1185505: The width of a rectangle is 2 less than twice its length. If the area of the rectangle is 181 cm2, what is the length of the diagonal?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let x equal the length.
then 2x - 2 equals the width.
the area of the rectangle is equal to x * (2x - 2) = 2x^2 - 2x which is equal to 181.
your quadratic equation is 2x^2 - 4x = 181
subtract 181 from both sides of that equation to get the standard quadratic form of 2x^2 - 4x - 181 = 0
factor that quadratic equation to get:
x = 10.026279441629 or x = -9.0262794416288
x has to be positive, so x = 10.026279441629
that's the length of the rectangle.
the width of the rectangle is equal to 2x - 2 which is equal to 18.05255888.
multiply length times the width to get an area of 181 which is correct.
the diagonal is equal to the square root of length squared plus width squared which is equal to the square root of (10.026279441629^2 + 18.05255888^2) which is equal to 20.6499676.
that's the length of the diagonal.
i did not do these calculations manually.
i used a quadratic equation solver that can be found on https://www.mathsisfun.com/quadratic-equation-solver.html
the manual calculations would have been horrendous.
the main thing to do was to get the length and the width modeled correctly.
if x is the length, and the width is 2 less than twice the length, then the width has to be 2x - 2.
since length * width = area, that gets you x * (2x - 2) = 181
simply to get 2x^2 - 2x = 181
subtract 181 from both sides of the equation to get:
2x^2 - 2x - 181 = 0
that's the input to the quadratic equation solver.
i used that when i found out that there was not going to be an easy solution for the roots.
finding the length of the diagonal was just an application of the pythagoren formula that tells you that the hypotenuse of a right triangle is equal to the square root of the square of the length plus the square of the width.
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