Find the volume of the largest right circular cylinder (in units^3) that fits in a sphere of radius 4 units.
To find the volume of the largest right circular cylinder that fits in a sphere, we need to make use of the properties of the sphere and cylinder.
First, let's identify the important dimensions of the problem. The sphere has a radius of 4 units, which means its diameter is 8 units. The cylinder will be inscribed within the sphere, so the diameter of the cylinder's circular base will also be 8 units.
Now, we need to determine the height of the cylinder. Since the cylinder should fit entirely inside the sphere, the height of the cylinder will be equal to the diameter of the sphere. Therefore, the height of the cylinder is 8 units.
To calculate the volume of the cylinder, we can use the formula: volume = π * radius^2 * height.
Since the diameter of the cylinder's circular base is 8 units, the radius is half of that, which is 4 units. The height of the cylinder is 8 units.
Therefore, the volume of the cylinder is calculated as follows:
volume = π * radius^2 * height
= π * 4^2 * 8
Simplifying the expression:
volume = π * 16 * 8
= 128π
Hence, the volume of the largest right circular cylinder that fits in a sphere with a radius of 4 units is 128π cubic units.
If you draw a side view, you can see that r^2+h^2 = 4^2
now maximize πr^2h = πh(16-h^2)