A conical shaped container, whose radius of base is r cm. and height is h cm., is full of water. A sphere of radius R is completely immersed in the container in such a way that the surface of sphere touches the base of the cone and its surfaces. The portion of water which comes out of the cone is.....

Options

A. `R^2/(r^2 h)`
B. `(r^2)/(R^2 h)`
C. `(4R^3)/(r^2 h)`
D. `(4r^2)/(R^2 h)`

Answer

To find the volume of water that comes out of a conical container when a sphere is immersed in it, we will follow these steps:

### Step 1: Determine the Volume of the Sphere
The volume \( V_s \) of a sphere is given by the formula:
\[ V_s = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the sphere.

### Step 2: Determine the Volume of the Cone
The volume \( V_c \) of a cone is given by the formula:
\[ V_c = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.

### Step 3: Calculate the Volume of Water Displaced
When the sphere is immersed in the cone, it displaces a volume of water equal to its own volume. Therefore, the volume of water that comes out of the cone is equal to the volume of the sphere:
\[ \text{Volume of water displaced} = V_s = \frac{4}{3} \pi R^3 \]

### Step 4: Write the Expression for the Volume of Water Displaced
The volume of water that comes out of the cone can be expressed as:
\[ \text{Volume of water that comes out} = \frac{4}{3} \pi R^3 \]

### Step 5: Find the Proportion of Water Displaced to Total Water Volume
To find the proportion of water that comes out, we divide the volume of water displaced by the total volume of water in the cone:
\[ \text{Proportion} = \frac{\text{Volume of water displaced}}{\text{Total volume of water in cone}} = \frac{\frac{4}{3} \pi R^3}{\frac{1}{3} \pi r^2 h} \]

### Step 6: Simplify the Expression
Now, we simplify the expression:
\[ \text{Proportion} = \frac{\frac{4}{3} \pi R^3}{\frac{1}{3} \pi r^2 h} = \frac{4R^3}{r^2 h} \]

### Final Answer
Thus, the portion of water that comes out of the cone is:
\[ \frac{4R^3}{r^2 h} \]