SOLUTION: The width of a rectangle is 6 less than twice its length. If the area of the rectangle is 113 cm^2, what is the length of the diagonal?

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Question 218423: The width of a rectangle is 6 less than twice its length. If the area of the rectangle is 113 cm^2, what is the length of the diagonal?
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
The width of a rectangle is 6 less than twice its length. If the area of the rectangle is 113 cm^2, what is the length of the diagonal?

Step 1. Let L be the length.

Step 2. Let 2L-6 be the width since the width of a rectangle is 6 less than twice its length.

Step 3. Area A=113=L*(2L-6)

Step 4. Solving for L yields the following steps

113=2L%5E2-6L

Subtract 113 from both sides of the equation to get a quadratic equation

113-113=2L%5E2-6L-113

2L%5E2-6L-113=0

To solve for L, we can use the quadratic formula given as

L+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=2, b=-6 and c=-113

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aL%5E2%2BbL%2Bc=0 (in our case 2L%5E2%2B-6L%2B-113+=+0) has the following solutons:

L%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=%28-6%29%5E2-4%2A2%2A-113=940.

Discriminant d=940 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--6%2B-sqrt%28+940+%29%29%2F2%5Ca.

L%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+940+%29%29%2F2%5C2+=+9.16485485837795
L%5B2%5D+=+%28-%28-6%29-sqrt%28+940+%29%29%2F2%5C2+=+-6.16485485837795

Quadratic expression 2L%5E2%2B-6L%2B-113 can be factored:
2L%5E2%2B-6L%2B-113+=+2%28L-9.16485485837795%29%2A%28L--6.16485485837795%29
Again, the answer is: 9.16485485837795, -6.16485485837795. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-6%2Ax%2B-113+%29



L=9.165 selecting the positive length
and
2L-6=12.33

Step 5. Let c be the diagonal.

Step 6. Use the Pythagorean Theorem to find the diagonal which says that the sum of the squares of the legs (length and width in our example) is equal to the sum of the squares of the hypotenuse (diagonal in this case). That is,

c%5E2=9.165%5E2%2B12.33%5E2=236.03

Take the square root to both sides of the equation.

sqrt%28c%5E2%29=sqrt%28236.03%29

c=15.36

Step 7. ANSWER: The diagonal is 15.36 cm.

I hope the above steps were helpful.

For FREE Step-By-Step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.

And good luck in your studies!

Respectfully,
Dr J