Find The altitude of a right circular cone of maximum curved surface which can be inscribed in a sphere of radius 'r'.

To find the altitude of a right circular cone of maximum curved surface that can be inscribed in a sphere of radius 'r', we first need to understand the relationship between the cone and the sphere.

Let's assume the altitude of the right circular cone is 'h', and the radius of the base of the cone is 'R'. We can consider the top of the cone as the apex, while the base of the cone lies on the surface of the sphere with a radius of 'r'.

Now, if we draw a line segment connecting the apex of the cone to the center of the sphere, we can form a right-angled triangle. The hypotenuse of this triangle is the radius of the sphere 'r', while the height of the triangle is the sum of the altitude of the cone 'h' and the radius of the cone's base 'R'.

Using the Pythagorean theorem, we can derive the relationship between 'r', 'h', and 'R':

r^2 = h^2 + R^2

To find the maximum curved surface area of the cone, we need to maximize the area of the curved surface, which is given by the formula:

A = πRℓ

where ℓ is the slant height of the cone. In this case, ℓ can be calculated by using the Pythagorean theorem:

ℓ^2 = h^2 + R^2

Since we are trying to maximize the surface area, we need to differentiate it with respect to both 'h' and 'R' and find the critical points using calculus. However, we already have an equation that relates 'h', 'R', and 'r', so we can substitute 'R' in terms of 'h' into the equation for surface area to get a single variable equation.

Substituting from our previous equation 'h^2 + R^2 = r^2':

ℓ^2 = h^2 + (r^2 - h^2) = r^2

Simplifying this equation, we get:

ℓ^2 = r^2

Taking the square root of both sides:

ℓ = r

Now, substituting this value of ℓ into the formula for the surface area:

A = πRℓ = πRh = πRh

Since we want to maximize the curved surface area, we have to maximize A, which means we want to maximize Rh.

As we can see, the value of ℓ does not depend on the height of the cone 'h'. Therefore, to maximize the curved surface area, we have to maximize the height of the cone 'h'. The maximum value of 'h' occurs when h = r.

So, the altitude of the right circular cone of maximum curved surface area that can be inscribed in a sphere of radius 'r' is equal to the radius of the sphere, which is 'r'.