If a right circular cylinder of height 10 is inscribed in a sphere of radius 6, what is the volume of the cylinder ?

Options

A. 104
B. 346
C. 545
D. 785

Explanation

To find the volume of the right circular cylinder inscribed in a sphere, we can follow these steps:

### Step 1: Understand the relationship between the cylinder and the sphere
We have a sphere with a radius of 6 units and a cylinder with a height of 10 units inscribed within it. The cylinder's radius will be determined using the Pythagorean theorem.

**Hint:** Remember that the height of the cylinder and the radius of the sphere form a right triangle with the radius of the cylinder.

### Step 2: Identify the dimensions
The height of the cylinder (h) is given as 10 units. The radius of the sphere (R) is 6 units. The radius of the cylinder (r) is what we need to find.

**Hint:** The radius of the cylinder can be found using the relationship between the height of the cylinder and the radius of the sphere.

### Step 3: Use the Pythagorean theorem
In the right triangle formed by the radius of the sphere, the radius of the cylinder, and half the height of the cylinder, we can express this relationship as:
\[ R^2 = r^2 + \left(\frac{h}{2}\right)^2 \]

Substituting the known values:
\[ 6^2 = r^2 + \left(\frac{10}{2}\right)^2 \]
\[ 36 = r^2 + 5^2 \]
\[ 36 = r^2 + 25 \]

**Hint:** Make sure to square the height divided by 2 correctly.

### Step 4: Solve for the radius of the cylinder
Rearranging the equation:
\[ r^2 = 36 - 25 \]
\[ r^2 = 11 \]

**Hint:** This gives you the square of the radius of the cylinder.

### Step 5: Calculate the volume of the cylinder
The volume (V) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]

Substituting the values we have:
\[ V = \pi (11) (10) \]
\[ V = 110\pi \]

**Hint:** You can approximate \(\pi\) as 3.14 or use the value of \(\pi\) directly if needed.

### Step 6: Approximate the volume
Using \(\pi \approx 3.14\):
\[ V \approx 110 \times 3.14 \]
\[ V \approx 346.4 \]

Thus, the volume of the cylinder is approximately 346 cubic units.

**Final Answer:** The volume of the cylinder is \( 110\pi \) or approximately 346 cubic units.