The heights of two cones are same and equal to 6cm .Their radii are 4cm and 3cm .They are melted and recast in to a cylinder of base radius 5cm.Find the height of this cylinder.

Answer

To solve the problem step by step, we need to find the height of the cylinder formed by melting two cones. Here’s how we can do it:

### Step 1: Calculate the Volume of the First Cone
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
For the first cone:
- Radius \( r_1 = 4 \) cm
- Height \( h = 6 \) cm

Substituting the values:
\[ V_1 = \frac{1}{3} \pi (4)^2 (6) = \frac{1}{3} \pi (16)(6) = \frac{96}{3} \pi = 32 \pi \text{ cm}^3 \]

### Step 2: Calculate the Volume of the Second Cone
Using the same formula for the second cone:
- Radius \( r_2 = 3 \) cm
- Height \( h = 6 \) cm

Substituting the values:
\[ V_2 = \frac{1}{3} \pi (3)^2 (6) = \frac{1}{3} \pi (9)(6) = \frac{54}{3} \pi = 18 \pi \text{ cm}^3 \]

### Step 3: Calculate the Total Volume of the Two Cones
Now, we add the volumes of both cones to get the total volume:
\[ V_{total} = V_1 + V_2 = 32 \pi + 18 \pi = 50 \pi \text{ cm}^3 \]

### Step 4: Set Up the Volume of the Cylinder
The volume of the cylinder \( V \) is given by:
\[ V = \pi r^2 h \]
Where:
- Radius \( r = 5 \) cm
- Height \( h \) is what we need to find.

Substituting the radius into the formula:
\[ V = \pi (5)^2 h = 25 \pi h \text{ cm}^3 \]

### Step 5: Equate the Volumes
Since the total volume of the melted cones is equal to the volume of the cylinder, we can set up the equation:
\[ 50 \pi = 25 \pi h \]

### Step 6: Solve for the Height \( h \)
To find \( h \), we can divide both sides by \( 25 \pi \):
\[ h = \frac{50 \pi}{25 \pi} = 2 \text{ cm} \]

### Conclusion
The height of the cylinder is \( 2 \) cm.

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