# Math Help - The rectangular playground quardratic equations question

1. ## The rectangular playground quardratic equations question

The question goes like this:

''A rectangular playground has a perimeter of $60m$ and an area of $216m^2$. Find the dimensions of the playground.''

What I did is this (perimeter):

Let the length be $x$ meters and the breadth be $y$ metres.
From the given information,
$2x + 2y = 60$

$2x +2y -60 = 0$

That was when I got stuck in this status quo...
It's common sense that I couldn't solve that equation.

Now if I were to take another way, which is the equation of the area of the playground:

$
(x^2)(y^2)= 216$

$x^2y^2 -216 = 0$

I still couldn't solve this...

Can anyone help me to solve this question? Thank you so much for taking your time!

2. You are correct in the first part of the equation:

$2x + 2y = 60m$

But your second equation is wrong. The area of a rectangle is base * height... so where did you get the x^2 and y^2 from?
The equation for the area should be:

$xy = 216m^2$

Now that we have 2 equations with 2 variables, we can use simultaneous equation to solve for x and y, the dimensions of the playground.

First equation: $xy = 216m^2$ which if we rewrite in the form of y= , we get $y = \frac{216}{x}$

Second equation: $2x + 2y = 60m$

Now we substitute from the first equation of y = 216/x into the second equation whenever y occurs. This leaves us with:

$2x + 2(\frac{216}{x}) = 60$

$2x + \frac{432}{x} = 60$

Multiply everything by x to make it easier to solve:
$2x^2 -60x + 432 = 0$

Now solve the equation for x, using the quadratic formula or by factorising. You should get 2 answers, and those are the dimensions of the playground.

3. Or, from 2x+ 2y= 60, x+ y= 30 so that y= 30- x. Putting that into xy= 316 gives the quadratic equation $x(30- x)= 30x- x^2= 316$ or $x^2- 30x+ 316= 0$. That is, of course, the same as $2x^2- 60x+ 432= 0$ divided by 2.

By the way, area is always measured in "square units" where "units" is whatever a distance is measured in. If x and y are measured in meters, for example, $x^2y^2$ would be $(m^2)(m^2)= m^4$, while $xy$ gives the correct $(meters)(meters)= meters^2$.