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 can be fully submerged as a result one third of the water in the original cone overflows what is the volume of each of the solid cone submerged

Let's solve this step by step.

### Problem Statement:
An inverted cone is filled with water to the brim. The cone has:
- Height \( h = 20 \) cm
- Diameter \( d = 16.8 \) cm, so the radius \( r = \frac{d}{2} = 8.4 \) cm

Two equal solid cones are dropped into this water-filled cone, causing one-third of the water to overflow. We need to find the volume of each of the solid cones.

### Step 1: Find the volume of the original cone filled with water.

The volume \( V \) of a cone is given by:

\[
V = \frac{1}{3} \pi r^2 h
\]

Substituting the given values:

\[
V = \frac{1}{3} \pi (8.4)^2 (20)
\]

First, calculate \( (8.4)^2 \):

\[
(8.4)^2 = 70.56
\]

Now, calculate the volume:

\[
V = \frac{1}{3} \pi \times 70.56 \times 20
\]

\[
V = \frac{1}{3} \times 1411.2 \pi
\]

\[
V = 470.4 \pi \text{ cubic cm}
\]

### Step 2: Determine the volume of water that overflows.

According to the problem, one-third of the water overflows:

\[
V_{\text{overflow}} = \frac{1}{3} \times 470.4 \pi
\]

\[
V_{\text{overflow}} = 156.8 \pi \text{ cubic cm}
\]

### Step 3: Determine the volume of each solid cone.

Since two equal cones cause this overflow, the volume of water displaced by each cone is half of the total overflow:

\[
V_{\text{cone}} = \frac{156.8 \pi}{2}
\]

\[
V_{\text{cone}} = 78.4 \pi \text{ cubic cm}
\]

### Final Answer:

The volume of each solid cone that is submerged is \( 78.4 \pi \) cubic cm. Numerically, this is approximately \( 246.31 \) cubic cm.

---

Would you like more details or have any questions? Here are five related questions to further your understanding:

1. If only one solid cone was dropped, how much water would overflow?
2. What would be the new height of the water if no water had overflowed but the cones were still submerged?
3. How does the shape of the submerged cone affect the volume of water displaced?
4. What would happen if the two cones had different volumes?
5. How would the overflow volume change if the cones were hollow instead of solid?

**Tip:** When dealing with the displacement of water by submerged objects, remember that the volume of the displaced water equals the volume of the submerged object.

Explore how to calculate the volume of two equal solid cones submerged in an inverted cone filled with water. Step-by-step solution includes finding the volume of the original cone, determining overflow, and calculating submerged cone volumes. Suitable for high school students learning about volumes of cones and water displacement.

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