SOLUTION: the length of a rectangle is 12 meters longer than half the width. the area of the rectangle is 90 square meters. find the dimensions of the rectangle.

Algebra ->  Rectangles -> SOLUTION: the length of a rectangle is 12 meters longer than half the width. the area of the rectangle is 90 square meters. find the dimensions of the rectangle.      Log On


   

Question 221567: the length of a rectangle is 12 meters longer than half the width. the area of the rectangle is 90 square meters. find the dimensions of the rectangle.
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
The length of a rectangle is 12 meters longer than half the width. The area of the rectangle is 90 square meters. Find the dimensions of the rectangle.

Step 1. Let w be the width.

Step 2. Let w%2F2%2B12 be the length.

Step 3. Area A=w%28w%2F2%2B12%29=90

Step 4. Solving A in Step 3 yields the following steps.

w%5E2%2F2+%2B12w=90

Multiply 2 to both sides to get rid of denominator

2w%5E2%2F2%2B2%2A12w=90%2A2

w%5E2%2B24w=180

Subtract 180 from both sides to get a quadratic equation

w%5E2%2B24w-180=180-180

w%5E2%2B24w-180=0

Step 5. To solve equation in Step 5, use the quadratic formula given below

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

where a=1, b=24, and c=-180

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aw%5E2%2Bbw%2Bc=0 (in our case 1w%5E2%2B24w%2B-180+=+0) has the following solutons:

w%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=%2824%29%5E2-4%2A1%2A-180=1296.

Discriminant d=1296 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-24%2B-sqrt%28+1296+%29%29%2F2%5Ca.

w%5B1%5D+=+%28-%2824%29%2Bsqrt%28+1296+%29%29%2F2%5C1+=+6
w%5B2%5D+=+%28-%2824%29-sqrt%28+1296+%29%29%2F2%5C1+=+-30

Quadratic expression 1w%5E2%2B24w%2B-180 can be factored:
1w%5E2%2B24w%2B-180+=+1%28w-6%29%2A%28w--30%29
Again, the answer is: 6, -30. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B24%2Ax%2B-180+%29



With w=6, then w%2F2%2B12=15. Area A=6*15=90 which is a true statement.

Step 6. ANSWER: The dimensions of a rectangle are 6 meters and 15 meters.

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.

Good luck in your studies!

Respectfully,
Dr J