Word Problem: Cone Insribed in Sphere

[guitarguy]

I have been able to do a few word problems concerning change in rates and in two dimensions. I am having trouble with this problem in three dimensions.

A right cone is inscribed in a sphere of radius a, find the maximum volume the cone can be in terms of radius a.

I know I can write x^2 + y^2 = a^2

And Volume of cone equals pi * r^2 * h.

I think the radius is x and the height is y+a but this yields an equation when differentiated makes no sense.

I need some kind of hint to help me get Volume of the Cone as a function of radius and then I can maximize the volume by differentiating.

Quite stuck, attempted many times.

 

[galactus]

Let the radius of the cone be r and the radius of the sphere be R.

The volume of the cone is \(\displaystyle \frac{\pi}{3}r^{2}h\)

But, \(\displaystyle h=R+\sqrt{R^{2}-r^{2}}\)

So, the cones volume is \(\displaystyle \frac{\pi}{2}r^{2}(R+\sqrt{R^{2}-r^{2}})\)

Differentiate w.r.t r, set to 0 and solve.

Try the First Derivative Test to test it is a maximum.

 

[HallsofIvy]

I have been able to do a few word problems concerning change in rates and in two dimensions. I am having trouble with this problem in three dimensions.

A right cone is inscribed in a sphere of radius a, find the maximum volume the cone can be in terms of radius a.

I know I can write x^2 + y^2 = a^2

No. That is the equation of a circle. A sphere with radius a and center at the origin is given by \(\displaystyle x^2+ y^2+ z^2= a^2\).

And Volume of cone equals pi * r^2 * h.

I think the radius is x and the height is y+a but this yields an equation when differentiated makes no sense.

No. The base of the cone is not on the xy-plane. I assume you are setting up the coordinate system so that the base of the cone is parallel to the xy-plane. If we take "\(\displaystyle z_0\)" to be the z coordinate of every point in the plane, then \(\displaystyle h= z_0+ a\). Further, that plane cuts the sphere where \(\displaystyle x^2+ y^2+ z_0^2= a^2\) which is the same as \(\displaystyle x^2+ y^2= a^2- z_0^2\) so that the radius of the base is \(\displaystyle r= a^2- z_0^2\). Then you can write the volume of the cone in terms of the single variable \(\displaystyle z_0\).

I need some kind of hint to help me get the volume of the Cone as a function of radius and then I can maximize the volume by differentiating.

Quite stuck, attempted many times.

 

[guitarguy]

Solution

Thanks for the help!

I was able to visualize the problem and calculate the volume as (32/81) * pi * a^3.

Which matches the answer in the book. I struggled a long time with this problem. My error was I had the plane of the cone in the x z plane when it is parallel to the x y plane!