Maximizing volume question where you must create the equation... Please help?

A sector of a circle with central angle theta is cut from a circle with radius R. The edges of teh sector are then glued to form a cone. Find the value of theta that produces a cone with maximum volume. Please show all work! NEEDS TO BE CALCULATED BY USING DERIVATIVES AND RELATIVE MAXIMUMS.

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1 Answer

  • 1 decade ago
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    For a circle of lenth 2R(pi), where pi is length of the circle divided by diameter.

    angle is 2(pi).

    To angle theta whats the lenght L ?

    Theta/2(pi) = L/ 2R(pi)

    L = R *Theta

    L is the lenght of the circular base of the cone.

    Volume of the cone is :

    V = (1/3)*Base area * height of the cone

    L is now the lenght of a new circle(radius r) :

    L = 2(pi)r = R * Theta

    r =R *Theta/2(pi)

    Base area : B =(pi) r² = (pi)R² * Theta²/4(pi²) = R² *Theta/4(pi)

    height: H = sqrt(R² - r²); R² because is the radius of the other circle !

    sqrt : square root

    V is maximum when dV/dTheta = 0

    Calculate it and you will find Theta.

    Theta = (pi)*sqrt(8/3) = 2*(pi)*[sqrt(2/3)]

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