Two metallic cones of equal radii 2.1 cm and heights 4.1 cm and 4.3 cm are metled together and recast a sphere. Find the radius of sphere so formed.

Answer

To find the radius of the sphere formed by melting two metallic cones, we will follow these steps:

### Step 1: Calculate the volume of each 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.

For the first cone:
- Radius \( r_1 = 2.1 \, \text{cm} \)
- Height \( h_1 = 4.1 \, \text{cm} \)

The volume \( V_1 \) of the first cone is:
\[ V_1 = \frac{1}{3} \pi (2.1)^2 (4.1) \]

For the second cone:
- Radius \( r_2 = 2.1 \, \text{cm} \) (same as the first cone)
- Height \( h_2 = 4.3 \, \text{cm} \)

The volume \( V_2 \) of the second cone is:
\[ V_2 = \frac{1}{3} \pi (2.1)^2 (4.3) \]

### Step 2: Combine the volumes of both cones
The total volume \( V_c \) of the two cones is:
\[ V_c = V_1 + V_2 \]
Substituting the volumes:
\[ V_c = \frac{1}{3} \pi (2.1)^2 (4.1) + \frac{1}{3} \pi (2.1)^2 (4.3) \]
Factoring out common terms:
\[ V_c = \frac{1}{3} \pi (2.1)^2 (4.1 + 4.3) \]
Calculating \( 4.1 + 4.3 = 8.4 \):
\[ V_c = \frac{1}{3} \pi (2.1)^2 (8.4) \]

### Step 3: Calculate the volume of the sphere
The volume \( V_s \) of the sphere is given by:
\[ V_s = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 4: Set the volumes equal
Since the volume of the melted cones equals the volume of the sphere:
\[ \frac{1}{3} \pi (2.1)^2 (8.4) = \frac{4}{3} \pi r^3 \]
We can cancel \( \frac{1}{3} \pi \) from both sides:
\[ (2.1)^2 (8.4) = 4 r^3 \]

### Step 5: Solve for \( r^3 \)
Calculating \( (2.1)^2 = 4.41 \):
\[ 4.41 \times 8.4 = 4 r^3 \]
Calculating \( 4.41 \times 8.4 = 37.044 \):
\[ 37.044 = 4 r^3 \]
Dividing both sides by 4:
\[ r^3 = \frac{37.044}{4} = 9.261 \]

### Step 6: Find \( r \)
Taking the cube root of both sides:
\[ r = \sqrt[3]{9.261} \approx 2.1 \, \text{cm} \]

### Final Answer
The radius of the sphere formed is approximately \( 2.1 \, \text{cm} \).

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