A right circular cone whose base radius is 4 is inscribed in a sphere of radius 5. what is the ratio of the volume of the cone to the volume of the sphere?

Options

A. 0.222:1
B. 0.256:1
C. 0.288:1
D. 0.333:1

Explanation

To find the ratio of the volume of the cone to the volume of the sphere, we will follow these steps:

### Step 1: Identify the given values
- Radius of the base of the cone, \( r_c = 4 \)
- Radius of the sphere, \( r_s = 5 \)

### Step 2: Set up the relationship using the Pythagorean theorem
The cone is inscribed in the sphere. We can visualize the situation as follows:
- The height of the cone is \( h \).
- The radius of the cone's base is \( r_c = 4 \).
- The radius of the sphere is \( r_s = 5 \).

We can form a right triangle where:
- The height of the cone from the base to the apex is \( h \).
- The radius of the cone's base is \( r_c = 4 \).
- The hypotenuse is the radius of the sphere \( r_s = 5 \).

Using the Pythagorean theorem:
\[ h^2 + r_c^2 = r_s^2 \]
Substituting the known values:
\[ h^2 + 4^2 = 5^2 \]
\[ h^2 + 16 = 25 \]
\[ h^2 = 25 - 16 \]
\[ h^2 = 9 \]
\[ h = 3 \]

### Step 3: Calculate the total height of the cone
Since the height \( h \) we found is the distance from the apex of the cone to the base, we need to consider the full height of the cone from the apex to the bottom of the sphere:
\[ \text{Total height of the cone} = r_s + h = 5 + 3 = 8 \]

### Step 4: Calculate the volume of the cone
The volume \( V_c \) of a cone is given by the formula:
\[ V_c = \frac{1}{3} \pi r_c^2 h \]
Substituting the values we have:
\[ V_c = \frac{1}{3} \pi (4^2)(8) = \frac{1}{3} \pi (16)(8) = \frac{128}{3} \pi \]

### Step 5: Calculate the volume of the sphere
The volume \( V_s \) of a sphere is given by the formula:
\[ V_s = \frac{4}{3} \pi r_s^3 \]
Substituting the value of the radius of the sphere:
\[ V_s = \frac{4}{3} \pi (5^3) = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \]

### Step 6: Calculate the ratio of the volumes
Now we find the ratio of the volume of the cone to the volume of the sphere:
\[ \text{Ratio} = \frac{V_c}{V_s} = \frac{\frac{128}{3} \pi}{\frac{500}{3} \pi} \]
The \( \pi \) and \( \frac{1}{3} \) cancel out:
\[ \text{Ratio} = \frac{128}{500} = \frac{64}{250} = \frac{32}{125} \]

### Final Answer
The ratio of the volume of the cone to the volume of the sphere is \( \frac{32}{125} \).