# Math Help - GIven Expression for Total Area

1. ## GIven Expression for Total Area

A piece of wire 20 m long is cut into two pieces, the length of the first piece being x m. The first piece is bent into a circle, and the other is bent into a rectangle with length twice the width. Give an expression for the total area A enclosed in the two shapes in terms of x.

MY WORK:

I divided 20m into x and (20 - x).

I then let TA = total area in terms of x.

I came up with the function

TA = pi(x)^2 + (40x - 2x^2)

Is this right?

2. Dividing the wire into pieces of length $x$ and $20-x$ is fine.

However, you should understand that these lengths serve as the circumference (length of the boundary of the circle) and the perimeter (length of the boundary of the rectangle). So, for example, $x$ is not the radius of the circle, and therefore your solution is wrong.

See if you can work the problem out with that in mind.

3. ## I...

Originally Posted by AlephZero
Dividing the wire into pieces of length $x$ and $20-x$ is fine.

However, you should understand that these lengths serve as the circumference (length of the boundary of the circle) and the perimeter (length of the boundary of the rectangle). So, for example, $x$ is not the radius of the circle, and therefore your solution is wrong.

See if you can work the problem out with that in mind.
I tried but couldn't figure out the function.

4. Originally Posted by magentarita
I tried but couldn't figure out the function.
Fair enough.... well for the circle portion of the question: We know that $x$ is the circumference, so $x=2\pi r$ from the formula for circumference. Solving for $r$, we have $r=x/2\pi.$ Now plug this value of $r$ into the formula for the area of the circle, and you should have your answer for the circle part in terms of $x$.

The rectangle portion of the question is solved in a similar way. See if you can figure out what it should be.

Let us know what answer you get.

5. ## I'll try later...

Originally Posted by AlephZero
Fair enough.... well for the circle portion of the question: We know that $x$ is the circumference, so $x=2\pi r$ from the formula for circumference. Solving for $r$, we have $r=x/2\pi.$ Now plug this value of $r$ into the formula for the area of the circle, and you should have your answer for the circle part in terms of $x$.

The rectangle portion of the question is solved in a similar way. See if you can figure out what it should be.

Let us know what answer you get.
I'll try to finish later.