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?
consider the sector subtended by an angle θ. The arc length to be removed is thus 9θ, which will be the circumference of the base of the cone. The base radius of the cone is thus
r = (18π - 9θ)/2π
Looking at a cross-section of the cone, we see that r^2 + h^2 = 9^2
The volume of the cone is
v = 1/3 π r^2 h
= 1/3 π r^2 √(81-r^2)
dv/dr = πr(54-r^2)/√(81-r^2)
dv/dr=0 when r^2 = 54, so r = 3√6
so, h^2 = 81-54 = 27
h = 3√3
Hmm. I wonder if h=R^(3/2) if the original circle is of radius R?
To find the height of the cone, we need to determine the size of the circular sector to maximize its volume.
Let's break the problem down into smaller steps:
Step 1: Determine the formula for the volume of a cone.
The volume of a cone is given by the formula V = (1/3) * π * r^2 * h, where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the radius of the base of the cone, and h is the height of the cone.
Step 2: Express the radius of the circular sector in terms of its angle.
To determine any angle α, we know that the length of the arc it spans is equal to the radius multiplied by the angle in radians: l = r * α. In this case, the length of the circular sector is equal to the circumference of the base of the cone, which is 2πr: 2πr = r * α. Simplifying the equation gives α = 2π.
Step 3: Express the radius of the circular sector in terms of its angle.
Now we can substitute the value of α into the previous equation: r * 2π = 9. Solve for r: r = 9 / 2π.
Step 4: Calculate the height of the cone.
We know that the height of the cone is the remaining length of the sector after the circle is removed. Since the length of the sector is 2πr, the height will be the radius minus the remaining length: h = 9 - (9 / 2π).
Step 5: Evaluate the height of the cone.
Finally, calculate the height of the cone: h ≈ 9 - (9 / 2 * 3.14159). Simplifying the equation gives the approximate value of h.
By following these steps, you can find the height of the cone.