SOLUTION: A rectangle is drawn so the width is 2 inches longer than the height. If the rectangle's diagonal measurement is 58 inches, find the height. Give your answer rounded to 1 decima

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Question 930748: A rectangle is drawn so the width is 2 inches longer than the height. If the rectangle's diagonal measurement is 58 inches, find the height.
Give your answer rounded to 1 decimal place.
Answer by rcdodds(6) About Me  (Show Source):
You can put this solution on YOUR website!
Well let's set this up as a system of equations.
Allow "w" to be the width.
Allow "h" to be the height.
Allow "d" to be the diagonal.
We know from the Pythagorean Theorem that...
a%5E2+%2B+b%5E2+=+c%5E2
but in this case
a=w
b=h
c=d=58, so we can say that...
w%5E2+%2B+h%5E2+=+d%5E2+=+58%5E2. Which is our first equation.
Our second equation comes from the "width is 2 inches longer than the height", which means that w+=+h+%2B+2.
Substituting that into the first equation we can solve using algebra to find the height.
w%5E2+%2B+h%5E2+=+58%5E2
%28h+%2B+2%29%5E2+%2B+h%5E2+=+58%5E2
h%5E2+%2B+4h+%2B+4+%2B+h%5E2+=+58%5E2
2h%5E2+%2B+4h+%2B+4+=+58%5E2
2h%5E2+%2B+4h+-+3360+=+0
h%5E2+%2B+2h+-+1680+=+0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ah%5E2%2Bbh%2Bc=0 (in our case 1h%5E2%2B2h%2B-1680+=+0) has the following solutons:

h%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%282%29%5E2-4%2A1%2A-1680=6724.

Discriminant d=6724 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+6724+%29%29%2F2%5Ca.

h%5B1%5D+=+%28-%282%29%2Bsqrt%28+6724+%29%29%2F2%5C1+=+40
h%5B2%5D+=+%28-%282%29-sqrt%28+6724+%29%29%2F2%5C1+=+-42

Quadratic expression 1h%5E2%2B2h%2B-1680 can be factored:
1h%5E2%2B2h%2B-1680+=+1%28h-40%29%2A%28h--42%29
Again, the answer is: 40, -42. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B2%2Ax%2B-1680+%29


Rounding this to one decimal place, we get our final answer to be 40 inches.
Note that we ignore the negative result from the quadratic formula because you can't have a negative distance.