588 ft of fencing what is x=? and y=?

Suppose that 588 ft of fencing are used to enclose a corral in the shape of a rectangle with a semicircle whose diameter is a side of the rectangle as the following figure

4 Answers

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  • 3 weeks ago

    You failed to give a figure. Let's supposed that x is the number of goats inside the fence, and y  is the number of apples you need to make an apple pie.  Then  x = 7 and y = 12.

  • Pope
    Lv 7
    3 weeks ago

    You have asked for x and y without defining either. Let me suppose that x is the diameter of the semicircle. That side of the corral must then be πx/2. The opposite side has length x. Let y be the length of each of the other two sides.

    x + πx/2 + 2y = 588 ft

    That is one linear equation with two unknowns. It has infinitely many solutions. Your question, such as it is, mentions a following figure, but never delivers. The answer may depend on something that is missing.

  • 3 weeks ago

    There's no figure.

    Let's assume that x is the length of two sides of the rectangle and the diameter of the (semi)circle, and y is the length of the other two sides of the rectangle, so we have:

    The circumference of the entire circle is pi*x, so the circumference of the semicircle is (pi/2)*x.

    The perimeter of the entire corral is:

    x + 2y + (pi/2)*x = 588

    2y = 588 - x - (pi/2)*x

    y = 294 - (x/2) - (pi/4)*x

    y = 294 - 0.5x - 0.25*pi*x

    The area of the rectangle is xy.

    The area of the entire circle is pi*(x/2)^2.

    The area of the semicircle is (pi/2)(x/2)^2 = 0.125*pi*x^2

    The area of the entire corral is:

    A = xy + 0.125*pi*x^2

    A = x(294 - 0.5x - 0.25*pi*x) + 0.125*pi*x^2

    A = 294x - 0.5x^2 - 0.25*pi*x^2 + 0.125*pi*x^2

    A = 294x - 0.5x^2 - 0.25*pi*x^2 + 0.125*pi*x^2

    A = 294x - 0.5x^2 - 0.125*pi*x^2

    Suppose you want to maximize the area of the corral, in which case let's find the derivative and set it to zero:

    dA/dx = 294 - x - 0.25*pi*x

    294 - x - 0.25*pi*x = 0

    x(1 + 0.25*pi) = 294

    x = 294 / (1 + 0.25*pi)

    x =~ 164.66915113239786851768071589287

    y = 294 - 0.5(294 / (1 + 0.25*pi)) - 0.25*pi*(294 / (1 + 0.25*pi))

    y = 147 / (1 + 0.25*pi)

    y =~ 82.334575566198934258840357946484

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