SOLUTION: the length of a rectangle Is 6 feet more than three times the width. the area of the rectanle is 240 square feet. what are the dimensions? 

Algebra ->  Rectangles -> SOLUTION: the length of a rectangle Is 6 feet more than three times the width. the area of the rectanle is 240 square feet. what are the dimensions?       Log On


   

Question 216320: the length of a rectangle Is 6 feet more than three times the width. the area of the rectanle is 240 square feet. what are the dimensions? 
Found 2 solutions by drj, Earlsdon:
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
The length of a rectangle is 6 feet more than three times the width. The area of the rectangle is 240 square feet. What are the dimensions?

Step 1. Let w be the width and 3w+6 is the length since rectangle is 6 feet more than three times the width.

Step 2. Area = width * length = w(3w+6) = 240 square feet

Step 3. Then we can put the equation in Step 2 in quadratic form

3w%5E2%2B6w=+240

Subtract 240 to both sides of the equation

3w%5E2%2B6w-240=0

Step 4. We can use the quadratic formula given as

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

where a=3, b=6, and c=-240

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aw%5E2%2Bbw%2Bc=0 (in our case 3w%5E2%2B6w%2B-240+=+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=%286%29%5E2-4%2A3%2A-240=2916.

Discriminant d=2916 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-6%2B-sqrt%28+2916+%29%29%2F2%5Ca.

w%5B1%5D+=+%28-%286%29%2Bsqrt%28+2916+%29%29%2F2%5C3+=+8
w%5B2%5D+=+%28-%286%29-sqrt%28+2916+%29%29%2F2%5C3+=+-10

Quadratic expression 3w%5E2%2B6w%2B-240 can be factored:
3w%5E2%2B6w%2B-240+=+3%28w-8%29%2A%28w--10%29
Again, the answer is: 8, -10. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B6%2Ax%2B-240+%29



So select the positive solution for positive dimensions:

w=8+ and 3w%2B6=3%2A8%2B6=30

Verify area.... 8%2A30=240...which is a true statement.

Step 5. Dimensions of the rectangle are a width of 8 feet and a length of 30 feet.

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

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Area = L*W = 240
L+=+3W%2B6 Substitute into the equation above:
%283W%2B6%29%2AW+=+240 Simplify.
3W%5E2%2B6W+=+240 Subtract 240 from both sides.
3W%5E2%2B6W-240+=+0 Factor a 3 just to simplify a bit.
3%28W%5E2%2B2W-80%29+=+0 so...
W%5E2%2B2W-80+=+0 Factor the trinomial.
%28W%2B10%29%28W-8%29+=+0 so that...
W%2B10+=+0 or W-8+=+0 which means that...
W+=+-10 or W+=+8 Discard the negative solution as width can only be a positive value.
W+=+8 and L+=+3%288%29%2B6
L+=+30
The length is 30 feet and the width is 8 feet.