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Math Help - GIven Expression for Total Area

  1. #1
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    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?

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  2. #2
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    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.
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  3. #3
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    I...

    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.
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  4. #4
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    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.
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  5. #5
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    I'll try later...

    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.
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