SOLUTION: The perimeter of a rectangle is 14cm. If the diagonal is 5cm, find the dimensions of the rectangle.

Algebra ->  Customizable Word Problem Solvers  -> Geometry -> SOLUTION: The perimeter of a rectangle is 14cm. If the diagonal is 5cm, find the dimensions of the rectangle.      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 949668: The perimeter of a rectangle is 14cm. If the diagonal is 5cm, find the dimensions of the rectangle.
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
The perimeter of a rectangle is 14cm. If the diagonal is 5cm, find the dimensions of the rectangle.
L=Length; W=width; D=Diagonal; P=perimeter=2(L+W)
D=sqrt%28L%5E2%2BW%5E2%29
5cm=sqrt%28L%5E2%2BW%5E2%29 square each side
25+sq+cm=%28L%5E2%2BW%5E2%29
P=2(L+W)
14 cm=2(L+W) Divide each side by 2.
7 cm=L+W Subtract W from each side
7 cm-W=L
25+sq+cm=%28L%5E2%2BW%5E2%29 Substitute for L
25+sq+cm=%287cm-W%29%5E2%2BW%5E2%29
25+sq+cm=W%5E2-14W%2B49%2BW%5E2 Subtract 25 cm from each side.
0=2W%5E2-14W%2B24cm
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aW%5E2%2BbW%2Bc=0 (in our case 2W%5E2%2B-14W%2B24+=+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=%28-14%29%5E2-4%2A2%2A24=4.

Discriminant d=4 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--14%2B-sqrt%28+4+%29%29%2F2%5Ca.

W%5B1%5D+=+%28-%28-14%29%2Bsqrt%28+4+%29%29%2F2%5C2+=+4
W%5B2%5D+=+%28-%28-14%29-sqrt%28+4+%29%29%2F2%5C2+=+3

Quadratic expression 2W%5E2%2B-14W%2B24 can be factored:
2W%5E2%2B-14W%2B24+=+2%28W-4%29%2A%28W-3%29
Again, the answer is: 4, 3. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-14%2Ax%2B24+%29

ANSWER The answers are 3 cm and 4 cm, so when width=3 cm, length=4 cm and when width=4 cm, length=3 cm.
CHECK
P=2(L+W)
14 cm=2(4 cm+3 cm)
14 cm=2(7 cm)
14 cm=14 cm
Diagonal=5
D=sqrt%28L%5E2%2BW%5E2%29
5+cm=sqrt%284%5E2%2B3%5E2%29
5+cm=sqrt%2816%2B9%29
5+cm=sqrt%2825+cm%5E2%29
5+cm=5+cm