A vessel, in the form of an inverted cone, is filled with water to the brim. Its height is 20 cm and diameter is 16.8 cm. Two equal solid cones are dropped in it so that they are fully submerged. As a result, one-third of the water in the original cone overflows. What is the volume of each of the solid cones submerged ?

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the given dimensions of the inverted cone.
- Height (h) = 20 cm
- Diameter (d) = 16.8 cm
- Radius (r) = d/2 = 16.8 cm / 2 = 8.4 cm

### Step 2: Calculate the volume of the inverted cone using the formula for the volume of a cone.
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values we have:
\[ V = \frac{1}{3} \times \frac{22}{7} \times (8.4)^2 \times 20 \]

### Step 3: Calculate \( (8.4)^2 \).
\[ (8.4)^2 = 70.56 \]

### Step 4: Substitute \( (8.4)^2 \) back into the volume formula.
\[ V = \frac{1}{3} \times \frac{22}{7} \times 70.56 \times 20 \]

### Step 5: Calculate \( \frac{22}{7} \times 70.56 \).
\[ \frac{22}{7} \times 70.56 \approx 220.8 \]

### Step 6: Substitute this value back into the volume formula.
\[ V = \frac{1}{3} \times 220.8 \times 20 \]

### Step 7: Calculate \( 220.8 \times 20 \).
\[ 220.8 \times 20 = 4416 \]

### Step 8: Divide by 3 to find the volume of the inverted cone.
\[ V = \frac{4416}{3} = 1472 \text{ cm}^3 \]

### Step 9: Since one-third of the water overflows when the two cones are submerged, calculate the volume of water that overflows.
\[ \text{Volume of water that overflows} = \frac{1}{3} \times 1472 = 490.67 \text{ cm}^3 \]

### Step 10: Since this volume is equal to the volume of the two submerged cones, we can find the volume of each cone.
\[ \text{Volume of each cone} = \frac{490.67}{2} = 245.33 \text{ cm}^3 \]

### Final Answer:
The volume of each of the solid cones submerged is approximately **245.33 cm³**.

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