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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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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