GIven Expression for Total Area

magentarita · #1 $Old$ July 16th, 2009, 05:25 PM

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?

AlephZero · #2 $Old$ July 16th, 2009, 05:57 PM

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.

magentarita · #3 $Old$ July 16th, 2009, 08:14 PM

Quote:

Originally Posted by AlephZero $View Post$

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.

AlephZero · #4 $Old$ July 16th, 2009, 08:25 PM

Quote:

Originally Posted by magentarita $View Post$

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.

magentarita · #5 $Old$ July 17th, 2009, 10:50 AM

Quote:

Originally Posted by AlephZero $View Post$

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.