Two solid spheres of radii 2 cm and 4 cm are melted and recast into a cone of height 8 cm. Find the radius of the cone so formed.

Answer

To solve the problem of finding the radius of the cone formed by melting two solid spheres, we will follow these steps:

### Step 1: Calculate the volume of the first sphere
The formula for the volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
For the first sphere with a radius of 2 cm:
\[ V_1 = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \, \text{cm}^3 \]

### Step 2: Calculate the volume of the second sphere
Using the same formula for the second sphere with a radius of 4 cm:
\[ V_2 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \, \text{cm}^3 \]

### Step 3: Find the total volume of the two spheres
Now, we add the volumes of the two spheres:
\[ V_{total} = V_1 + V_2 = \frac{32}{3} \pi + \frac{256}{3} \pi = \frac{288}{3} \pi = 96 \pi \, \text{cm}^3 \]

### Step 4: Set the total volume equal to the volume of the cone
The volume of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( h \) is the height of the cone. Given that the height \( h = 8 \) cm, we can write:
\[ V_{cone} = \frac{1}{3} \pi r^2 (8) = \frac{8}{3} \pi r^2 \]

### Step 5: Equate the volumes
Setting the total volume of the spheres equal to the volume of the cone:
\[ 96 \pi = \frac{8}{3} \pi r^2 \]

### Step 6: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ 96 = \frac{8}{3} r^2 \]
Multiplying both sides by 3 to eliminate the fraction:
\[ 288 = 8 r^2 \]

### Step 7: Solve for \( r^2 \)
Now, divide both sides by 8:
\[ r^2 = \frac{288}{8} = 36 \]

### Step 8: Find \( r \)
Taking the square root of both sides gives:
\[ r = \sqrt{36} = 6 \, \text{cm} \]

Thus, the radius of the cone formed is **6 cm**.