Find the radius of the base and altitude of a right circular cone of maximum volume that could be inscribed in a sphere at radius 10 meters.
see related questions below. And
http://math.stackexchange.com/questions/225718/find-the-volume-of-the-largest-right-circular-cone-that-can-be-inscribed-in-a-sp
To find the radius of the base and the altitude of a right circular cone of maximum volume inscribed in a sphere, we can follow these steps:
Step 1: Visualize the problem
Imagine a sphere with a radius of 10 meters. Inside this sphere, a cone is inscribed. The cone's base will be a circle, and its apex (or peak) will touch the sphere's surface. We need to find the radius of the base and the altitude (height) of this cone.
Step 2: Understand the geometric properties
In a right circular cone, the radius of the base and the height are related to each other. We can use the Pythagorean theorem to express this relationship. Let's denote the cone's radius of the base as r and its height as h.
Step 3: Relate the cone and the sphere
Because the cone is inscribed in the sphere, the diameter of the sphere must be equal to the diagonal of the cone's base. The diagonal can be expressed as the square root of the sum of the square of the radius and the square of the height. Therefore, the diameter of the sphere is equal to 2r, which means r is equal to half the diameter of the sphere.
Step 4: Calculate the maximum cone volume
The volume of a cone can be calculated using the formula: V = (π * r^2 * h) / 3. We want to maximize the volume, so we need to express it only in terms of one variable.
Step 5: Express the volume equation in terms of r
Substituting the value of h from the Pythagorean theorem, we get h = sqrt(100 - r^2). Now we can substitute h back into the volume equation to get V = (π * r^2 * sqrt(100 - r^2)) / 3.
Step 6: Find the derivative of the volume equation
We need to find the derivative of the volume equation with respect to r, which will help us determine where the volume is maximized. Differentiating the volume equation, we get dV/dr = (2π * r * sqrt(100 - r^2) - (π * r^3) / sqrt(100 - r^2)) / 3.
Step 7: Set the derivative equal to zero and solve for r
To find the maximum volume, we set the derivative equal to zero and solve the equation: (2π * r * sqrt(100 - r^2) - (π * r^3) / sqrt(100 - r^2)) / 3 = 0.
Step 8: Solve for r
Solving the above equation for r is slightly complicated. We can first multiply both sides of the equation by 3 * sqrt(100 - r^2), then square both sides to eliminate the square roots. After simplification, the resulting equation can be solved using algebraic techniques such as factoring or the quadratic formula.
Step 9: Calculate h using the value of r
Once you have found the value of r, you can substitute it back into the Pythagorean theorem to calculate the height (h) of the cone.
By following these steps, you can find the radius of the base and the altitude of a right circular cone of maximum volume that can be inscribed in a sphere with a radius of 10 meters.