SOLUTION: The length of a rectangle is 5 less than thrice its width. Find the dimensions if the area is 112 squared cm.

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Question 154298: The length of a rectangle is 5 less than thrice its width. Find the dimensions if the area is 112 squared cm.
Found 2 solutions by jojo14344, Alan3354:
Answer by jojo14344(1513) About Me  (Show Source):
You can put this solution on YOUR website!
Remember:
Arect=L%2AW -------------------------------------> working eqn
But L=3W-5, 5 less thrice the width
So substitute,
A=%283W-5%29%28W%29, distribute
112=%283w%5E2-5W%29
3W%5E2-5W-112=0
By Quadratic: a=3, b=-5, & c=-112
x=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29
x=%28-%28-5%29%2B-sqrt%28-5%5E2-4%2A3%2A-112%29%29%2F%282%2A3%29
x=%285%2B-sqrt%2825%2B1344%29%29%2F6
x=%285%2B-sqrt%281369%29%29%2F6
x=%285%2B-37%29%2F6, It has 2 values:
x=%285%2B37%29%2F6=7 --------------------> TO BE USED , WIDTH
x=%285-37%29%2F6=-5.33 -----------------> DON'T USED (-)
Therefore,
L=3%2A7-5=21-5=16cm
W=7cm
In doubt? Go back working eqn:
A=L%2AW
112=16%2A7
112cm%5E2=112cm%5E2
Thank you,
Jojo

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The length of a rectangle is 5 less than thrice its width. Find the dimensions if the area is 112 squared cm.
---------------------------
L = 3W - 5 (The length of a rectangle is 5 less than thrice its width)
L*W = 112
Sub for L
(3W-5)*W = 112
3w^2 - 5W - 112 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B-5x%2B-112+=+0) has the following solutons:

x%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=%28-5%29%5E2-4%2A3%2A-112=1369.

Discriminant d=1369 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--5%2B-sqrt%28+1369+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+1369+%29%29%2F2%5C3+=+7
x%5B2%5D+=+%28-%28-5%29-sqrt%28+1369+%29%29%2F2%5C3+=+-5.33333333333333

Quadratic expression 3x%5E2%2B-5x%2B-112 can be factored:
3x%5E2%2B-5x%2B-112+=+%28x-7%29%2A%28x--5.33333333333333%29
Again, the answer is: 7, -5.33333333333333. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B-5%2Ax%2B-112+%29

Ignore the negative answer, so W = 7 (the online solver always uses x)
W = 7
L = 3*7 - 5 = 16