What size "slice of pie" should be removed from a circle of radius R?

This is an optimization problem. The volume V of a cone is (1/3)*A*h, where A is the area of the base and h is the height. The area of the base will be π*r^2, where r is the radius of the base. So V = (1/3)π*(r^2)*h. Depending on the slice taken, r will vary. You won't know r directly, but you will know the circumference of the base; if angle θ is cut out of the circle, the remaining circumference is R*(2π - θ). Since the circumference is 2πr, r will be R*(2π - θ) / (2π), which can also be written as R*(1 - (θ/2π)). The height h of the cone will also be a little tricky to get. The slant height of the cone will always be R, and you just found the radius r of the base. By the pythagorean theorem, the height h will be sqrt(R^2 - r^2). So now you have an expression for the volume of the cone in terms of r and h, an expression for h with r as the only unknown, and an expression for r with θ as the only unknown. Thus, you can write an expression for V with θ as the only unknown. Once you've done that, you can use any optimization method you like; your best bet is to differentiate with respect to θ and then solve for θ where the derivative is equal to zero.

al of it by a piece has the maximun volume