SOLUTION: If the perimeter of a rectangle is p and its diagnoal is d, the difference between the length and width of the rectangle is ______? I have worked several formulas, 2L Plus 2W =

Algebra ->  Rectangles -> SOLUTION: If the perimeter of a rectangle is p and its diagnoal is d, the difference between the length and width of the rectangle is ______? I have worked several formulas, 2L Plus 2W =       Log On


   

Question 892582: If the perimeter of a rectangle is p and its diagnoal is d, the difference between the length and width of the rectangle is ______?
I have worked several formulas, 2L Plus 2W = p, solved for L, and tryig to solve for W, I have worked it several ways. I have also used the Pathagoreum theorm, L squared plus w squared = d squared. I have not been able to solve it, begining to wonder if it is a trick problem from instructor. Can you help.
Answer by josgarithmetic(39799) About Me  (Show Source):
You can put this solution on YOUR website!
This can only be done using variables because no values are given.

p=2w%2B2L and w%5E2%2BL%5E2=d%5E2;
You want to find abs%28w-L%29.
You assume that the constants are d and p.

2L=p-2w
L=%28p-2w%29%2F2;
substitute into the diagonal equation.
w%5E2%2B%28p-2w%29%5E2%2F4=d%5E2
4w%5E2%2B%28p-2w%29%5E2=4d%5E2
4w%5E2%2Bp%5E2-4pw%2B4w%5E2-d%5E2=0
8w%5E2-4pw%2Bp%5E2-d%5E2=0
Use the general solution of quadratic equation.
w=%284p%2B-+sqrt%2816p%5E2-4%2A8%28p%5E2-d%5E2%29%29%29%2F%2816%29
w=%284p%2B-+sqrt%2816p%5E2-32%28p%5E2-d%5E2%29%29%29%2F16
w=%284p%2B-+4sqrt%28p%5E2-2%28p%5E2-d%5E2%29%29%2F16
w=%28p%2B-+sqrt%28p%5E2-2p%5E2%2B2d%5E2%29%29%2F4
highlight_green%28w=%28p%2B-+sqrt%282d%5E2-p%5E2%29%29%2F4%29-----this is just for w.

What about the other dimension, L?
... w=(p-2L)/2;
%28p-2L%29%5E2%2F4%2BL%5E2=d%5E2 when substituted into the diagonal equation.
%28p-2L%29%5E2%2B4L%5E2=4d%5E2
p%5E2-4pL%2B4L%5E2%2B4L%5E2=d%5E2
p%5E2-4pL%2B8L%5E2-d%5E2=0
8L%5E2-4pL%2Bp%5E2-d%5E2=0
General Solution,
L=%284p%2B-+sqrt%2816p%5E2-4%2A8%28p%5E2-d%5E2%29%29%29%2F16
L=%284p%2B-+sqrt%2816p%5E2-16%2A2%28p%5E2-d%5E2%29%29%29%2F16
L=%284p%2B-+4sqrt%28p%5E2-2%28p%5E2-d%5E2%29%29%2F16
L=%28p%2B-+sqrt%28p%5E2-2p%5E2%2B2d%5E2%29%29%2F4
highlight_green%28L=%28p%2B-+sqrt%282d%5E2-p%5E2%29%29%2F4%29-----formula for L.

YOU can finish this.
Pay attention to both forms for w and L, since each has a PLUS form and a MINUS form. Remember, you are looking for THE DIFFERENCE or absolute value of w and L.

Please excuse the lack of rendering in three of the steps. Tracing the parentheses will be difficult. The majority of readable steps should be plenty helpful.