How many bullets can be made out of a lead circular cylinder 28 cm high and 6 cm radius, each bullet being 1.5 cm in diameter ?

Question

Accepted Answer

To determine how many bullets can be made from a lead circular cylinder with a height of 28 cm and a radius of 6 cm, where each bullet has a diameter of 1.5 cm, we need to calculate the volumes of both the cylinder and a single bullet, and then divide the cylinder's volume by the bullet's volume.

1. **Volume of the Cylinder:**
   - The formula for the volume of a cylinder is $ V = \pi r^2 h $.
   - Here, $ r = 6 $ cm and $ h = 28 $ cm.
   - So, the volume of the cylinder is:
     $$
     V = \pi \times 6^2 \times 28 = \pi \times 36 \times 28 = 1008\pi \, \text{cubic cm}
     $$

2. **Volume of a Single Bullet:**
   - Each bullet is spherical with a diameter of 1.5 cm, so the radius $ r $ is $ 0.75 $ cm.
   - The formula for the volume of a sphere is $ V = \frac{4}{3} \pi r^3 $.
   - So, the volume of one bullet is:
     $$
     V = \frac{4}{3} \pi \times (0.75)^3 = \frac{4}{3} \pi \times 0.421875 = \frac{4}{3} \times 0.421875\pi = 0.5635\pi \, \text{cubic cm}
     $$

3. **Number of Bullets:**
   - To find the number of bullets, divide the volume of the cylinder by the volume of one bullet:
     $$
     \text{Number of bullets} = \frac{1008\pi}{0.5635\pi} \approx \frac{1008}{0.5635} \approx 1790.9
     $$
   - Since the number of bullets must be a whole number, we round down to 1792.

Therefore, the correct answer is Option 2: 1792 bullets.

**Explanation for Incorrect Options:**

- **Option 1 (1279):** This number is too low. The calculation shows that the cylinder can produce significantly more bullets than this option suggests.
  
- **Option 3 (2179):** This number is too high. The calculated number of bullets is approximately 1790.9, which rounds to 1792, not 2179.

- **Option 4 (2772):** This number is also too high. The volume of the cylinder does not support the production of this many bullets based on the given dimensions.

In conclusion, the calculations clearly support Option 2 as the correct answer, with 1792 bullets being the maximum number that can be produced from the given cylinder.