A cylinder of height 4 cm and base radius 3 cm is melted to form a sphere. The radius of sphere is:

Options

A. 3 cm
B. `3.5` cm
C. 4 cm
D. 2.5 cm

Answer

To find the radius of the sphere formed by melting a cylinder, we need to equate the volumes of the cylinder and the sphere. Here’s how we can solve the problem step-by-step:

### Step 1: Calculate the Volume of the Cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height.

Given:
- Height \( h = 4 \) cm
- Base radius \( r = 3 \) cm

Substituting the values into the formula:
\[ V = \pi (3)^2 (4) = \pi \times 9 \times 4 = 36\pi \text{ cm}^3 \]

### Step 2: Set the Volume of the Sphere Equal to the Volume of the Cylinder
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the sphere.

Since the cylinder is melted to form the sphere, we can set the volumes equal to each other:
\[ 36\pi = \frac{4}{3} \pi R^3 \]

### Step 3: Cancel \( \pi \) from Both Sides
Dividing both sides by \( \pi \):
\[ 36 = \frac{4}{3} R^3 \]

### Step 4: Solve for \( R^3 \)
To eliminate the fraction, multiply both sides by \( 3 \):
\[ 108 = 4R^3 \]

Now, divide both sides by \( 4 \):
\[ R^3 = \frac{108}{4} = 27 \]

### Step 5: Calculate the Radius \( R \)
To find \( R \), take the cube root of both sides:
\[ R = \sqrt[3]{27} = 3 \text{ cm} \]

Thus, the radius of the sphere is **3 cm**.

### Final Answer
The radius of the sphere is **3 cm**.

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