A solid metallic right cone is melted and some solid right circular cylinders are made. If the radius of base of each cylinder is half the radius of cone and the height of each cylinder is one third the height of the cone, then the number of cylinders formed are.

Options

A. `3`
B. `4`
C. `5`
D. `6`

Answer

To solve the problem, we need to find out how many solid right circular cylinders can be formed from a solid metallic right cone when the cone is melted. Let's break it down step by step.

### Step 1: Write the formula for the volume of the cone.
The volume \( V \) of a right circular cone is given by the formula:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base of the cone and \( h \) is the height of the cone.

### Step 2: Write the formula for the volume of the cylinder.
The volume \( V \) of a right circular cylinder is given by the formula:
\[ V_{cylinder} = \pi r_{cylinder}^2 h_{cylinder} \]
where \( r_{cylinder} \) is the radius of the base of the cylinder and \( h_{cylinder} \) is the height of the cylinder.

### Step 3: Substitute the values for the cylinder's dimensions.
According to the problem:
- The radius of the base of each cylinder is half the radius of the cone, so:
\[ r_{cylinder} = \frac{r}{2} \]
- The height of each cylinder is one third the height of the cone, so:
\[ h_{cylinder} = \frac{h}{3} \]

### Step 4: Substitute these values into the volume formula for the cylinder.
Now, substituting \( r_{cylinder} \) and \( h_{cylinder} \) into the volume formula for the cylinder:
\[ V_{cylinder} = \pi \left(\frac{r}{2}\right)^2 \left(\frac{h}{3}\right) \]
Calculating this gives:
\[ V_{cylinder} = \pi \left(\frac{r^2}{4}\right) \left(\frac{h}{3}\right) = \frac{\pi r^2 h}{12} \]

### Step 5: Set the volume of the cone equal to the total volume of the cylinders.
Let \( n \) be the number of cylinders formed. The total volume of the cylinders is:
\[ n \cdot V_{cylinder} = n \cdot \frac{\pi r^2 h}{12} \]
Setting this equal to the volume of the cone:
\[ \frac{1}{3} \pi r^2 h = n \cdot \frac{\pi r^2 h}{12} \]

### Step 6: Cancel out common terms.
We can cancel \( \pi r^2 h \) from both sides (assuming \( r \) and \( h \) are not zero):
\[ \frac{1}{3} = n \cdot \frac{1}{12} \]

### Step 7: Solve for \( n \).
Multiplying both sides by 12 gives:
\[ 4 = n \]
Thus, the number of cylinders formed is \( n = 4 \).

### Final Answer:
The number of cylinders formed is **4**.

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