A vessel, in the form of an inverted cone, is filled with water to the brim. Its height is 32 cm and diameter of the base is 25.2 cm. Six equal solid cones are dropped in it, so that they are fully submerged. As a result, one-fourth of water in the original cone overflows. What is the volume of each of the solid cones submerged ?

Answer

To find the volume of each solid cone submerged in the inverted cone vessel, we can follow these steps:

### Step 1: Calculate the volume of the original cone
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.

Given:
- Height \( h = 32 \) cm
- Diameter of the base \( d = 25.2 \) cm, thus the radius \( r = \frac{d}{2} = \frac{25.2}{2} = 12.6 \) cm

Now, substituting the values into the formula:
\[ V = \frac{1}{3} \pi (12.6)^2 (32) \]

### Step 2: Calculate \( (12.6)^2 \)
\[ (12.6)^2 = 158.76 \]

### Step 3: Substitute back into the volume formula
\[ V = \frac{1}{3} \pi (158.76) (32) \]

### Step 4: Calculate \( V \) using \( \pi \approx \frac{22}{7} \)
\[ V = \frac{1}{3} \cdot \frac{22}{7} \cdot 158.76 \cdot 32 \]

### Step 5: Simplify the calculation
First, calculate \( \frac{158.76 \cdot 32}{3} \):
\[ 158.76 \cdot 32 = 5079.36 \]
\[ \frac{5079.36}{3} = 1693.12 \]

Now, substituting this back:
\[ V = \frac{22}{7} \cdot 1693.12 \]

### Step 6: Calculate the volume
\[ V \approx 22 \cdot 241.88 \approx 5321.36 \text{ cm}^3 \]

### Step 7: Calculate the volume of water that overflows
Since one-fourth of the water overflows:
\[ \text{Volume of water that overflows} = \frac{1}{4} V = \frac{1}{4} \cdot 5321.36 = 1330.34 \text{ cm}^3 \]

### Step 8: Calculate the volume of each solid cone
Since six equal solid cones are submerged:
\[ \text{Volume of each solid cone} = \frac{1330.34}{6} \approx 221.72 \text{ cm}^3 \]

### Final Answer
The volume of each solid cone submerged is approximately:
\[ \text{Volume of each solid cone} \approx 221.72 \text{ cm}^3 \]

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