SOLUTION: I need help with this question: The area of a rectangular rug is 160sqft. if it's width is 2ft less than 1/2 it's length, find the dimensions of the rug. I know A=LW. I'm racking

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Question 964189: I need help with this question: The area of a rectangular rug is 160sqft. if it's width is 2ft less than 1/2 it's length, find the dimensions of the rug.
I know A=LW. I'm racking my brain, but this is not my area of expertise. The equation I wrote to find the W is W=L/2-2. then i wrote 160=L(L/2-2), but that isn't getting me anywhere. I need help figuring out how to write the equation to find the dimensions. PLEASE HELP! :-)
Found 2 solutions by josgarithmetic, Alan3354:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Try to work with mostly lower cased letters for the variables.

w, width
L, length (upper case so not to confuse with any digit 1).
A, area of the rug

You already know, A=L%2Aw.
Translate the description into symbolism.

A=160;
wL=160;
w=-2%2B%281%2F2%29L.

What do you have?
Basically two equations in the two unknown variables, w and L. Use simple substitution.

system%28wL=160%2Cw=L%2F2-2%29

%28L%2F2-2%29L=160
At this point you should no longer be "racking" your brain, because you now have a quadratic equation in one single variable, L. You already found this far and then asked for help.
highlight_green%28L%5E2%2F2-2L=160%29
2%28L%5E2%2F2-2L%29=2%2A160
L%5E2-4L=320
highlight_green%28L%5E2-4L-320=0%29
-
Not even trying to factor that, ....
discriminant is %28-4%29%5E2-4%2A1%2A%28-320%29=16%2B1280=highlight_green%281296=%2836%29%5E2%29.
-
General solution for quadratic equation:
L=%284%2B-+sqrt%2836%5E2%29%29%2F2
L=%284%2B-+36%29%2F2
Obviously the PLUS-form makes sense only.
L=%284%2B36%29%2F2
L=40%2F2
highlight%28L=20%29.

Use either the area equation OR the description's formula equation to find w.
wL=160
w=160%2FL
w=160%2F20
highlight%28w=8%29

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The area of a rectangular rug is 160sqft. if it's [sic] width is 2ft less than 1/2 it's [sic] length, find the dimensions of the rug.
I know A=LW. I'm racking my brain, but this is not my area of expertise. The equation I wrote to find the W is W=L/2-2. then i wrote 160=L(L/2-2)
===================
160=L(L/2-2)
Your eqn is correct, just solve it.
320 = L*(L-4)
L^2 - 4L - 320 = 0
(L - 20)*(L + 16) = 0
Ignore the L = -16
---
L = 20
W = 8
==================
PS it's = it is