SOLUTION: the sum of the length and the width of a rectangle is 14. Find the values of the width for which the area is AT LEAST 45. clearly write algebraic expressions, this is not guess-and

Algebra ->  Rectangles -> SOLUTION: the sum of the length and the width of a rectangle is 14. Find the values of the width for which the area is AT LEAST 45. clearly write algebraic expressions, this is not guess-and      Log On


   

Question 1049776: the sum of the length and the width of a rectangle is 14. Find the values of the width for which the area is AT LEAST 45. clearly write algebraic expressions, this is not guess-and-check.
Answer by advanced_Learner(501) About Me  (Show Source):
You can put this solution on YOUR website!
l+w=14
area=lw
area=(14-w)*w

(14-w)*w>=45
14w-w^2>=45
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aw%5E2%2Bbw%2Bc=0 (in our case -1w%5E2%2B14w%2B-45+=+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=%2814%29%5E2-4%2A-1%2A-45=16.

Discriminant d=16 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-14%2B-sqrt%28+16+%29%29%2F2%5Ca.

w%5B1%5D+=+%28-%2814%29%2Bsqrt%28+16+%29%29%2F2%5C-1+=+5
w%5B2%5D+=+%28-%2814%29-sqrt%28+16+%29%29%2F2%5C-1+=+9

Quadratic expression -1w%5E2%2B14w%2B-45 can be factored:
-1w%5E2%2B14w%2B-45+=+-1%28w-5%29%2A%28w-9%29
Again, the answer is: 5, 9. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B14%2Ax%2B-45+%29