SOLUTION: section 9.6 # 46 Geometry. The length of a rectangle is 1 cm longer than its width. If the diagonal of the rectangle is 5 cm, what are the dimensions (the length and width) of

Algebra ->  Rectangles -> SOLUTION: section 9.6 # 46 Geometry. The length of a rectangle is 1 cm longer than its width. If the diagonal of the rectangle is 5 cm, what are the dimensions (the length and width) of      Log On


   

Question 73993: section 9.6 # 46

Geometry. The length of a rectangle is 1 cm longer than its width. If the diagonal of the rectangle is 5 cm, what are the dimensions (the length and width) of the rectangle?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let L=length, w=width.
The length and width can be represented as
L=w%2B1
The diagonal of the rectangle is the hypotenuse of the triangle with legs of L and w. So the length and width can be found by Pythagoreans theorem.
%28L%29%5E2%2Bw%5E2=5%5E2Plug in w+1 into L
%28w%2B1%29%5E2%2Bw%5E2=25foil the (w+1)^2 term
w%5E2%2B2w%2B2%2Bw%5E2=25Combine like terms and get everything to one side
2w%5E2%2B2w-24=0
Plug this into the quadratic equation to solve for w
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aw%5E2%2Bbw%2Bc=0 (in our case 2w%5E2%2B2w%2B-24+=+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=%282%29%5E2-4%2A2%2A-24=196.

Discriminant d=196 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+196+%29%29%2F2%5Ca.

w%5B1%5D+=+%28-%282%29%2Bsqrt%28+196+%29%29%2F2%5C2+=+3
w%5B2%5D+=+%28-%282%29-sqrt%28+196+%29%29%2F2%5C2+=+-4

Quadratic expression 2w%5E2%2B2w%2B-24 can be factored:
2w%5E2%2B2w%2B-24+=+2%28w-3%29%2A%28w--4%29
Again, the answer is: 3, -4. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B2%2Ax%2B-24+%29

This means the width is 3 (the negative width is ignored since it's not practical). So the length is
L=w%2B1
L=3%2B1
L=4
So the dimensions are: Width=3,Length=4


Check:
3%5E2%2B4%5E2=5%5E2
9%2B16=25
25=25Works