SOLUTION: The area of a rectangle is 270 squared cm. If the shorter side was reduced by 2cm and the longer side was increased by 4cm then the area would increase by 16 squared cm. Find the l
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Question 521326: The area of a rectangle is 270 squared cm. If the shorter side was reduced by 2cm and the longer side was increased by 4cm then the area would increase by 16 squared cm. Find the length of the sides of the original rectangle. (Thank you) Found 2 solutions by richwmiller, RiiRiiDaDragon:Answer by richwmiller(17219) (Show Source):
You can put this solution on YOUR website! Hey the previous tutor gave you the correct answers but the steps are totally missing... So in case you're still left in the dark i'll give you my 2 cents
Okay!
So lets call the longer side "x" and the shorter side "y"
Regardless of which is length or width the formula is the same.
Area = length x width
Therefore Area = xy
The question says that the original area is 270cm^2
So xy = 270cm^2
*NB
Keep in mind, we can manipulate this equation as we have a like depending on what we need so
x = 270\y
and
y = 270\x
Get it? Got it? Good
Alright moving on!!
So the question tells you to increase the longer side (x) by 4cm and to decrease the shorter side (y) by 2 cm and the 'new' area will increase by 16cm^2
So in a form we can actually use, this basically means:
(x+4)(y-2)=270+16
(x+4)(y-2)=286
Alright so we'll expand this (i'm gonna assume that you know how to expand)and then we'll get
xy-2x+4y-8=286
Still with me? Alright kl
So in order to work this with a quadratic formula we need to change some stuff around
Remember back when i said that although 270 = xy we can alter it as we see fit to get what we need?
Yeah .. if you don't i suggest scrolling back up cause this is where it comes in
(^.^)
x = 270\y
and
y = 270\x
For this equation i'm gonna write y in terms of x which is this thingamajiggy below:
y = 270\x
and inject it into the equation
Stay with me
so instead of this:
xy-2x+4y-8=286
I get this:
x(270\x)-2x+4(270\x)-8=286
We have to simplify that first and get
270-2x+1080\x)-8=286
And then put everything as a fraction of x( as in make x the denominator by multiplying everything on the left side of the equation except '1080" by x
So we're left with
(270x-2x^2+1080-8x)\x = 286
Alright. I always hated fractions... ugh so lets get rid of the x denominator by multiplying both sides by x. This cancels out the x denominator and multiplies the 286 by x on the left side of the equation.
270x-2x^2+1080-8x= 286x
Let's switch things up a bit so it looks more like a quadratic equation
-2x^2+270x-8x+1080=286x
All quadratic equation need to be equal to zero before we can implement the formula so subtract 286x from both sides
-2x^2+270x-8x+1080=286x-286x
-2x^2+270x-8x+1080-286x=0
That gives us
-2x^2-24x+1080=0
So FINALLY we can use the quadratic formula to solve the equation
Quadratic equation (in our case ) has the following solutons:
For these solutions to exist, the discriminant should not be a negative number.
First, we need to compute the discriminant : .
Discriminant d=9216 is greater than zero. That means that there are two solutions: .
Quadratic expression can be factored:
Again, the answer is: -30, 18.
Here's your graph:
Whether you used the quadratic solver i so lovingly offered or did it by hand or calculator or prayer, your answers should be
x=18 or x= -30
Distance can't be negative so the -30 is useless to us.
Now what we focus on is the POSITIVE 18!!
So with x=18
We draw for the old equation
xy=270
Now that we know what x is its now
18y=270
y=270\18
y=15
REALLY HOPE THIS HELPED... I KNOW ITS LONG BUT IT SHOULD GET THE JOB DONE
=)
