A metallic sphere of radus 4 cm and a metallic cone of base readius 3 cm and height 6 cm are melted and recast a cylinder of 5 cm radus. Find the height of the cylinder.

Answer

To find the height of the cylinder formed by melting a metallic sphere and a metallic cone, we will follow these steps:

### Step 1: Calculate the Volume of the Sphere
The formula for the volume of a sphere is given by:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Here, the radius \( r \) of the sphere is 4 cm. Substituting the value:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \text{ cm}^3 \]

### Step 2: Calculate the Volume of the Cone
The formula for the volume of a cone is given by:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Here, the radius \( r \) of the cone is 3 cm and the height \( h \) is 6 cm. Substituting the values:
\[ V_{\text{cone}} = \frac{1}{3} \pi (3)^2 (6) = \frac{1}{3} \pi (9)(6) = \frac{54}{3} \pi = 18 \pi \text{ cm}^3 \]

### Step 3: Calculate the Total Volume of the Sphere and Cone
Now, we add the volumes of the sphere and the cone:
\[ V_{\text{total}} = V_{\text{sphere}} + V_{\text{cone}} = \frac{256}{3} \pi + 18 \pi \]
To add these, we convert \( 18 \pi \) to have a common denominator:
\[ 18 \pi = \frac{54}{3} \pi \]
Thus,
\[ V_{\text{total}} = \frac{256}{3} \pi + \frac{54}{3} \pi = \frac{310}{3} \pi \text{ cm}^3 \]

### Step 4: Calculate the Volume of the Cylinder
The formula for the volume of a cylinder is given by:
\[ V_{\text{cylinder}} = \pi r^2 h \]
Here, the radius \( r \) of the cylinder is 5 cm. We can express the volume of the cylinder as:
\[ V_{\text{cylinder}} = \pi (5)^2 h = 25 \pi h \]

### Step 5: Set the Volumes Equal to Each Other
Since the total volume from the sphere and cone is equal to the volume of the cylinder, we have:
\[ \frac{310}{3} \pi = 25 \pi h \]
Dividing both sides by \( \pi \):
\[ \frac{310}{3} = 25h \]

### Step 6: Solve for \( h \)
To find \( h \), we rearrange the equation:
\[ h = \frac{310}{3 \times 25} = \frac{310}{75} \]
Now simplifying:
\[ h = \frac{62}{15} \approx 4.13 \text{ cm} \]

### Final Answer
The height of the cylinder is approximately \( 4.13 \) cm.

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