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 cone of greatest volume that can be made, we need to optimize the volume function. The volume of a cone can be calculated using the formula V = (1/3) * π * r^2 * h, where V is the volume, π is a constant (approximately 3.14), r is the radius, and h is the height.
Let's start by finding the relationships between the variables in this problem. We know that the base of the cone is a circle, and the radius of this circle is R inches.
When the sector is removed from the circular piece, the remaining part will be in the shape of a sector of a smaller circle. Let's assume the smaller circle has a radius of r inches.
To find the height of the cone, we need to calculate the length of the slant height (hypotenuse) of the sector. The slant height can be found using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the hypotenuse is the slant height, and the other two sides are r and R. Therefore, we have:
R^2 = r^2 + h^2
To simplify the problem, let's express h in terms of r:
h^2 = R^2 - r^2
h = sqrt(R^2 - r^2)
Now, we can substitute the value of h in the volume formula to get an expression for the volume in terms of r:
V = (1/3) * π * r^2 * sqrt(R^2 - r^2)
To find the radius, height, and volume of the cone of greatest volume, we need to optimize this volume function. To do that, we will take the derivative of the volume function with respect to r, set it equal to zero, and solve for r.
Let's differentiate the volume function with respect to r:
dV/dr = (1/3) * π * (2r * sqrt(R^2 - r^2) + r^2 * (-1/(2 * sqrt(R^2 - r^2))))
Now, let's set dV/dr equal to zero and solve for r:
0 = (1/3) * π * (2r * sqrt(R^2 - r^2) + r^2 * (-1/(2 * sqrt(R^2 - r^2))))
Simplifying the equation, we get:
2r * sqrt(R^2 - r^2) - (r^2 / (2 * sqrt(R^2 - r^2))) = 0
To solve this equation, we can multiply through by 2 * sqrt(R^2 - r^2) to get rid of the denominators:
2r * (R^2 - r^2) - r^2 = 0
Expanding and rearranging the equation:
2rR^2 - 2r^3 - r^2 = 0
Now, let's solve for r. Factor out an r:
r(2R^2 - 2r^2 - 1) = 0
From this equation, we find that either r = 0 or 2R^2 - 2r^2 - 1 = 0.
If r = 0, then the cone has no volume and is not of interest to us.
Therefore, we need to solve the equation 2R^2 - 2r^2 - 1 = 0.
Simplifying, we get:
2r^2 = 2R^2 - 1
Dividing through by 2, we have:
r^2 = R^2 - 1/2
Taking the square root of both sides, we find:
r = sqrt(R^2 - 1/2)
Now that we have the value of r, we can substitute it into the equation for h:
h = sqrt(R^2 - r^2)
h = sqrt(R^2 - (R^2 - 1/2))
h = sqrt(1/2)
Therefore, the height of the cone is sqrt(1/2) inches.
Now, substituting the values of r and h into the volume formula, we can find the volume:
V = (1/3) * π * r^2 * h
V = (1/3) * π * (sqrt(R^2 - 1/2))^2 * sqrt(1/2)
V = (π / 3) * (R^2 - 1/2) * sqrt(1/2)
Simplifying further, we get:
V = (π / 6) * (R^2 - 1/2) * sqrt(2)
Therefore, the radius of the cone is sqrt(R^2 - 1/2) inches, the height is sqrt(1/2) inches, and the volume is (π / 6) * (R^2 - 1/2) * sqrt(2) cubic inches.