Maximize Volume of Right Circular Cone with Constant Slant Edge

[disfused_3289]

1. The slant edge of a right circular cone is 6 cm in length. Find the height of the cone when the volume is a maximum.

2. Find the maximum volume of a right circular cone whose slant edge has a constant length measure a.

 

[dodo]

I would add,
3. Express the volume V(x) of a right circular cone in terms of the length x of its slant edge.

Then solve 3, then 2, then 1.

 

[tacman]

1. To find the height of the cone when the volume is a maximum, we can use the formula for the volume of a right circular cone: V = (1/3)πr^2h, where r is the radius of the base and h is the height. Since the slant edge is 6 cm, we can use the Pythagorean theorem to find the radius: r^2 + h^2 = 6^2. We can also express r in terms of h by using the formula for the slant height of a right circular cone: l = √(r^2 + h^2). Substituting this into the previous equation, we get h^2 + (√(r^2 + h^2))^2 = 6^2. Simplifying, we get h^2 + r^2 + h^2 = 36. Rearranging, we get 2h^2 + r^2 = 36.

To maximize the volume, we need to find the value of h that will give us the largest possible value for V. To do this, we can take the derivative of V with respect to h and set it equal to 0: dV/dh = (1/3)πr^2(2h) = (2/3)πr^2h = 0. Since r is a constant, we can ignore it and focus on the term (2/3)πh. Setting this equal to 0, we get h = 0. This is not a valid solution since the height of a cone cannot be 0. However, this does tell us that when h = 0, the volume is at a minimum.

To find the maximum volume, we can use the second derivative test by taking the derivative of dV/dh: d^2V/dh^2 = (2/3)πr^2. Since r is a constant, this is always positive, meaning that the volume is concave up and we have a maximum at h = 0. Therefore, the height of the cone when the volume is a maximum is 0 cm.

2. To find the maximum volume of a right circular cone with a constant slant edge of length a, we can use the same process as above, but this time we will keep a as a variable. The formula for the slant height of a cone is