Question

A rectangle's length `y` is half the square of it's width `x`. The perimeter is 48m. What are the rectangle's dimensions?

Answer 1

`y=18, x=6`

Explanation:

We have a rectangle with length, `y`, and width, `x`.

We are told two things about this rectangle:

• the length, `y`, is half the square of the width, `x`. We can write that as `y=1/2 x^2`

• the perimeter is 48m. The formula for the perimeter of a rectangle is `P=2y+2x`

We now have two equations and two variables, so we can solve it. I'm first going to substitute the `y=1/2 x^2` equation into the `P=2y+2x` equation (and replace `P` with 48):

`48=2(1/2 x^2)+2x`

And now let's solve for `x`:

`x^2+2x-48=0`

`(x+8)(x-6)=0`

`x=-8, 6`

Since we can't have negative length, `x=6`

We can now put this value of `x` into one of the original equations to get the dimension `y`:

`y=1/2 x^2`

`y=1/2 (6^2)`

`y=1/2 (36)`

`y=18`