SOLUTION: A rectangle and a square have the same area. The width of the rectangle is 96cm less than the side of the square and the length of the rectangle is 256cm more than twice the side o

Algebra ->  Rectangles -> SOLUTION: A rectangle and a square have the same area. The width of the rectangle is 96cm less than the side of the square and the length of the rectangle is 256cm more than twice the side o      Log On


   

Question 1085081: A rectangle and a square have the same area. The width of the rectangle is 96cm less than the side of the square and the length of the rectangle is 256cm more than twice the side of the square. What are the dimensions of the rectangle
Found 2 solutions by Fombitz, Theo:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The area of the square is,
A%5Bs%5D=s%5E2
For the rectangle,
W=s-96
L=256%2B2s
A=L%2AW
s%5E2=%28s-96%29%282s%2B256%29
s%5E2=2s%5E2%2B64s-24576
s%5E2%2B64s-24576=0
Complete the square,
s%5E2%2B64s%2B1024-24576=1024
%28s%2B32%29%5E2=24576%2B1024
%28s%2B32%29%5E2=25600
s%2B32=0+%2B-+160
Lengths are never negative.
s=128cm
So then the dimensions of the rectangle are,
W=128-96
W=32cm
and
L=256%2B2%28128%29
L=512cm

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
S is the size of the square.
L is the size of the length of the rectangle.
W is the size of the width of the rectangle.
area of the rectangle are the same.
S^2 = L * W

the width of the rectangle is 96 cm less than the side of the square.
W = S - 96

the length of the rectangle is 256 cm more than twice the side of the square.
L = 2 * S + 256

since W = S - 96 and L = 2S + 256, then S^2 = L * W becomes:

(S - 96) * (2 * S + 256) = S^2

simplify by multiplying out the factors to get:

2 * S^2 + 256 * S - 96 * 2 * S - 96 * 256 = S^2

simplify to get:

2 * S^2 + 256 * S - 192 * S - 24576 = S^2

combine like terms to get:

2 * S^2 + 64 * S - 24576 = S^2

subtract s^2 from both sides of the equation to get:

S^2 + 64 * S - 24576 = 0

factor the quadratic to get:

(S - 128) * (S + 192) = 0

solve for S to get:

S = 128 or S = -192

S can't be negative, so S = 128.

area of the square is 128 * 128 = 16384

W = 128 - 96 which makes W = 32

L = 2 * 128 + 256 which makes L = 512.

area of the rectangle is 32 * 512 = 16384

the area of the square is the same as the area of the rectangle.
the width of the rectangle is 96 less than the side of the square.
the length of the rectangle is 256 more than twice the side of the square.
all requirements of the problem are satisfied.
the solution looks good.

the solution is that the width of the rectangle is equal to 32 and the length of the rectangle is equal to 512.

in the process, you discovered that the side of the square is equal to 128.