SOLUTION: If the AREA of a rectangle is 20m and the PERIMETER of the rectangle is 20m, what are the length and the width?

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Question 214050: If the AREA of a rectangle is 20m and the PERIMETER of the rectangle is 20m, what are the length and the width?
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
If the AREA of a rectangle is 20 m^2 and the PERIMETER of the rectangle is 20 m, what are the length and the width?

Step 1. Area A=20=l*w where l is the length and w is the width for a rectangle.

Then l=20/w

Step 2. Perimeter P=20=l+l+2+2=2(l+w) where perimeter means adding up all four sides of a rectangle.

Step 3. Substitute the length l of Step 1 into Step 2 to get the following:

P=20=2%28l%2Bw%29

20=2%2820%2Fw%2Bw%29

Simplify and multiply by w to both sides of equation to get rid of denominator

10=20%2Fw%2Bw

w%2A10=w%2A%2820%2Fw%2Bw%29

10w=20%2Bw%5E2

Subtract 10w to both sides of equation to get a quadratic equation.


w%5E2-10w%2B20=0

where we can use the quadratic formula x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=1 b=-10 and c=20

Step 4. The following steps will solve the quadratic equation.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-10x%2B20+=+0) has the following solutons:

x%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=%28-10%29%5E2-4%2A1%2A20=20.

Discriminant d=20 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--10%2B-sqrt%28+20+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-10%29%2Bsqrt%28+20+%29%29%2F2%5C1+=+7.23606797749979
x%5B2%5D+=+%28-%28-10%29-sqrt%28+20+%29%29%2F2%5C1+=+2.76393202250021

Quadratic expression 1x%5E2%2B-10x%2B20 can be factored:
1x%5E2%2B-10x%2B20+=+1%28x-7.23606797749979%29%2A%28x-2.76393202250021%29
Again, the answer is: 7.23606797749979, 2.76393202250021. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-10%2Ax%2B20+%29



Step 5. ANSWER: Based on the above results, we have the width w=7.2361 m and the length l=2.7639 m. The numbers can also be reversed and note that these numbers satisfy the given area and perimeter.

So the sides of the rectangle are 7.2361 m and 2.7639 m.


I hope the above steps were helpful.

For free Step-By-Step Videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra or for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.

And good luck in your studies!

Respectfully,
Dr J