A conical drinking cup is made from a circular piece of paper of radius R inches by cutting out a sector and joining the edges CA and CB. Find the radius, height and volume of the cone of greatest volume that can be made this way
To find the radius, height, and volume of the cone of greatest volume that can be made from the circular piece of paper, we can use calculus and optimization techniques. Let's break down the problem into steps:
Step 1: Determine the dimensions of the sector to be cut out.
Let the angle of the sector be α. Since the circular piece of the paper has a radius of R, the perimeter of the sector is 2πR. Since the edges CA and CB are joined to create the cone, the circumference of the base of the cone is 2πR - CA - CB. And since the base of the cone is a circle of radius r, the circumference can be written as 2πr. Therefore, we can set up the equation:
2πR - αR - αR = 2πr
Simplifying, we get:
2πR - 2αR = 2πr
(π - α)R = πr
Therefore, αR = (π - α)r
Step 2: Express the volume of the cone as a function of α.
The volume of a cone is given by the formula V = (1/3)πr²h, where r is the radius and h is the height. Since r = R - α and the height h is equal to CA = CB, we can write the volume as:
V = (1/3)π(R - α)²h
V = (1/3)π(R² - 2Rα + α²)h
Step 3: Maximize the volume.
To maximize the volume of the cone, we need to find the value of α that maximizes V. This can be done by taking the derivative of V with respect to α, setting it equal to zero, and solving for α.
dV/dα = (1/3)π(-2Rh + 2αh)
Setting dV/dα = 0, we get:
-2R + 2α = 0
2α = 2R
α = R
Step 4: Find the radius, height, and volume.
Now that we have α = R, we can substitute this value back into our equations to find the radius, height, and volume:
r = R - α = R - R = 0
h = CA = CB = 2R
V = (1/3)π(R - α)²h = (1/3)π(0)²(2R) = 0
Therefore, the radius of the cone is 0 inches, the height is 2R inches, and the volume is 0 cubic inches. This means that the cone of greatest volume that can be made from the circular piece of paper has no actual volume since it doesn't exist.