A right circular cylinder whose diameter is equal to its height, is inscribed in a right circular cone of base diameter 16 cm and height 3 times the base diameter. The axes of both solids coincide. What is the volume (in `cm^(3)` ) of the solid inside the cone but outside the cylinder?

Options

A. 296 `pi`
B. 512 `pi`
C. 432 `pi`
D. 592 `pi`

Answer

To solve the problem, we need to find the volume of the solid inside the cone but outside the inscribed cylinder. We will follow these steps:

### Step 1: Determine the dimensions of the cone
The diameter of the cone's base is given as 16 cm. Therefore, the radius \( r_c \) of the cone is:
\[ r_c = \frac{16}{2} = 8 \text{ cm} \]
The height \( h_c \) of the cone is given as three times the base diameter:
\[ h_c = 3 \times 16 = 48 \text{ cm} \]

### Step 2: Calculate the volume of the cone
The formula for the volume \( V_c \) of a cone is:
\[ V_c = \frac{1}{3} \pi r^2 h \]
Substituting the values of \( r_c \) and \( h_c \):
\[ V_c = \frac{1}{3} \pi (8)^2 (48) = \frac{1}{3} \pi (64)(48) \]
Calculating this gives:
\[ V_c = \frac{1}{3} \pi (3072) = 1024 \pi \text{ cm}^3 \]

### Step 3: Determine the dimensions of the cylinder
The problem states that the diameter of the cylinder is equal to its height. Let the radius of the cylinder be \( r \) and its height be \( h \). Thus:
\[ h = 2r \]

### Step 4: Use similar triangles to find the radius of the cylinder
The triangles formed by the cone and the cylinder are similar. The height of the cone is 48 cm, and the height of the cylinder is \( h = 2r \). The remaining height of the cone above the cylinder is:
\[ 48 - h = 48 - 2r \]
Using the similarity of triangles:
\[ \frac{r}{8 - r} = \frac{2r}{48} \]
Cross-multiplying gives:
\[ 48r = 2r(8 - r) \]
Expanding and rearranging:
\[ 48r = 16r - 2r^2 \implies 2r^2 - 32r = 0 \implies 2r(r - 16) = 0 \]
Thus, \( r = 0 \) or \( r = 16 \). Since \( r \) must be less than 8 cm (the radius of the cone), we discard \( r = 16 \) and find:
\[ r = 6 \text{ cm} \]
Then, the height of the cylinder is:
\[ h = 2r = 2 \times 6 = 12 \text{ cm} \]

### Step 5: Calculate the volume of the cylinder
The volume \( V_{cy} \) of the cylinder is given by:
\[ V_{cy} = \pi r^2 h \]
Substituting the values:
\[ V_{cy} = \pi (6)^2 (12) = \pi (36)(12) = 432 \pi \text{ cm}^3 \]

### Step 6: Calculate the volume of the solid inside the cone but outside the cylinder
To find the volume of the solid inside the cone but outside the cylinder, we subtract the volume of the cylinder from the volume of the cone:
\[ V = V_c - V_{cy} = 1024 \pi - 432 \pi = 592 \pi \text{ cm}^3 \]

### Final Answer
The volume of the solid inside the cone but outside the cylinder is:
\[ \boxed{592 \pi \text{ cm}^3} \]