A solid metallic cylinder with a base radius of 3 cm and a height 120 of cm is melted and recast into a sphere of radius r cm. What is the value of r if there is a `10%` metal loss in the conversion

Options

A. 9
B. 8
C. `13.5`
D. 12

Answer

To find the radius \( r \) of the sphere formed from the melted cylinder, we will follow these steps:

### 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 of the cylinder
- \( h \) is the height of the cylinder

Given:
- Base radius \( r = 3 \) cm
- Height \( h = 120 \) cm

Substituting the values:

\[ V = \pi (3^2)(120) = \pi (9)(120) = 1080\pi \text{ cm}^3 \]

### Step 2: Account for the 10% metal loss
Since there is a 10% loss in metal during the conversion, we need to calculate the effective volume of the metal that can be used to form the sphere.

The volume after the loss is:

\[ \text{Volume after loss} = V - 0.1V = 0.9V \]

Substituting the volume of the cylinder:

\[ \text{Volume after loss} = 0.9 \times 1080\pi = 972\pi \text{ cm}^3 \]

### Step 3: Set up the equation for the volume of the sphere
The formula for the volume \( V \) of a sphere is given by:

\[ V = \frac{4}{3} \pi r^3 \]

We set this equal to the volume after the loss:

\[ \frac{4}{3} \pi r^3 = 972\pi \]

### Step 4: Solve for \( r^3 \)
We can cancel \( \pi \) from both sides:

\[ \frac{4}{3} r^3 = 972 \]

Now, multiply both sides by \( \frac{3}{4} \):

\[ r^3 = 972 \times \frac{3}{4} = 729 \]

### Step 5: Calculate \( r \)
To find \( r \), we take the cube root of both sides:

\[ r = \sqrt[3]{729} = 9 \text{ cm} \]

Thus, the radius \( r \) of the sphere is \( 9 \) cm.

### Final Answer
The value of \( r \) is \( 9 \) cm.

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