A paper cone is to be formed by starting with a disk of radius 9cm, cutting out a circular sector, and gluing the new edges together. The size of the circular sector is chosen to maximize the volume of the resulting cone. How tall is the cone?
To find the height of the cone that maximizes its volume, we need to set up an equation and then find the critical points. Let's go through the steps:
1. Start with a disk of radius 9 cm. The circumference of this disk is 2πr, where r is the radius. So, the circumference of the disk is 2π(9) = 18π cm.
2. Cut out a circular sector from the disk. Let's assume the angle of this sector is θ. The arc length of this sector is given by (θ/360) × circumference, which becomes (θ/360) × (18π) = (θ/20)π cm.
3. Now, we glue the new edges together to form a cone. The resulting cone has a base circumference equal to the arc length of the circular sector, which is (θ/20)π cm.
4. Let's assume the height of the cone is h cm.
5. The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
6. We know that the radius of the cone is 9 cm and the base circumference is (θ/20)π cm. The radius can be calculated using the formula C = 2πr, so r = (θ/40) cm.
7. Substituting the values of r and h into the volume formula, we get V = (1/3)π[(θ/40)^2]h = (πθ^2h)/4800 cm^3.
To find the value of θ that maximizes the volume V, we need to differentiate the volume equation with respect to θ. So, dV/dθ = (2πθh)/4800.
Setting dV/dθ equal to zero and solving for θ, we get:
(2πθh)/4800 = 0
⇒ 2πθh = 0
⇒ θ = 0 or h = 0
Since we cannot have a cone with a height of 0, we ignore the possibility of h = 0.
Therefore, θ = 0, which means no circular sector is cut out from the disk. In this case, the resulting cone would be fully formed as a disk with no height.
So, the height of the cone is 0 cm if we want to maximize its volume by cutting out a circular sector from the disk.