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Thread: Maximizing and Minimizing Area

  1. #1
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    Post Maximizing and Minimizing Area

    A wire 4 meters long is cut into two pieces. One piece is bent into a square for a frame for a stained glass ornament, while the other piece is bent into a circle for a TV antenna. To reduce storage space, where should the wire be cut to minimize the total area of both figures? Give the length of wire used for each:
    For the square:
    For the circle:

    Where should the wire be cut to maximize the total area?
    For the square:
    For the circle:

    I'm completely stumped. Can someone help me?
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    Quote Originally Posted by iheartthemusic29 View Post
    A wire 4 meters long is cut into two pieces. One piece is bent into a square for a frame for a stained glass ornament, while the other piece is bent into a circle for a TV antenna. To reduce storage space, where should the wire be cut to minimize the total area of both figures? Give the length of wire used for each:
    For the square:
    For the circle:

    Where should the wire be cut to maximize the total area?
    For the square:
    For the circle:

    I'm completely stumped. Can someone help me?
    let $\displaystyle x$ = length of wire for the circle
    $\displaystyle 4-x$ = length of wire for the square

    $\displaystyle 2\pi r = x$

    $\displaystyle r = \frac{x}{2\pi}$

    circle area, $\displaystyle A_c = \frac{x^2}{4\pi}$

    side length of square is $\displaystyle \frac{4-x}{4}$

    square area, $\displaystyle A_s = \frac{(4-x)^2}{16}$

    total area, $\displaystyle A = A_c + A_s$ ... find $\displaystyle \frac{dA}{dx}$ and minimize
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