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.
If you need a picture I can send one through email.
- 1 decade agoFavorite Answer
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)]