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

1 Answer

#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#