Find the dimension of the right circular cylinder of the largest volume that can inscribed in a
Sphere of radius 10 units.
To find the dimensions of the right circular cylinder with the largest volume that can be inscribed in a sphere of radius 10 units, we can apply some geometric properties.
Step 1: Understand the problem.
We need to find the dimensions of a cylinder. The cylinder is inscribed in a sphere, which means the sphere's surface contains the entire cylinder. The sphere has a radius of 10 units.
Step 2: Visualize the problem.
Imagine a sphere with a radius of 10 units. We need to fit a cylinder inside this sphere such that the cylinder's height is along the diameter of the sphere.
Step 3: Analyze the problem.
We know that the diameter of the sphere is equal to twice the radius, which means the diameter is 20 units. Since the height of the cylinder is along the diameter, it will have the same length as the sphere's diameter, which is 20 units.
Step 4: Calculate the dimensions.
The dimensions of the cylinder are as follows:
- Height (h): 20 units
- Radius (r): To find the cylinder's radius, we need to determine how much of the cylinder's height fits within the sphere. Since the cylinder height matches the diameter of the sphere, the radius is half the height: r = 20/2 = 10 units.
So, the dimensions of the right circular cylinder with the largest volume inscribed in the sphere with a radius of 10 units are:
- Height (h): 20 units
- Radius (r): 10 units
T = angle up from center of sphere to intersection of cylinder and sphere
r = 10
height of cylinder = 2 r sin T
radius of cylinder = r cos T
volume of cylinder =
V = pi(r^2 cos^2 T)(2 r sin T)
= 2pi r^2 (cos^2 T sin T)
dV/dT = 0 for max
= cos^3 T - 2 sin^2T cos T
= cosT ( cos^2 T - 2 sin^2 T
so for max
2 sin^2 T = cos^2 T
sin^2 T/cos^2 T = tan^2 T = 1/2
or
tan T = sqrt 2/2
T = 35.3 degrees3
height = 2 r sin T = 11.6
radius = r cos T = 8.16