Bob and Jack play a shooting game. In this
game, they have a net at a distance. The net is
made up of 1 mm thick wires. The wires run
vertically and horizontally to form a mesh that
has 5 mm square openings between them. A
bullet 2 mm in diameter must be shot in such a
way that it goes through the net without
touching the wires. Such a shot would result in
100 points. All other scenarios will result in 0
points. Bob is set to start the game. What is
the probability that Bob scores 100 points
from his first shot?
To calculate the probability that Bob scores 100 points from his first shot, we need to determine the area of the net where the bullet can pass through without touching the wires, and divide it by the total area of the net.
Let's break down the problem step by step:
1. Calculate the area of the net where the bullet can pass through:
- The wires form a mesh with 5 mm square openings between them.
- Since the bullet has a diameter of 2 mm, we can imagine a circle being inscribed within each square opening.
- The area of each circle is πr², where r is the radius of the circle.
- Given that the diameter of the bullet is 2 mm, the radius is 1 mm.
- Therefore, the area of each circle is π(1 mm)² = π mm².
- Since there are 25 (5x5) square openings in total, the total area of the circles is 25π mm².
2. Calculate the total area of the net:
- The dimensions of the net are not provided. However, we can assume a hypothetical size for calculation purposes.
- Let's assume the net is a 1000 mm x 1000 mm square.
- In this case, the total area of the net is (1000 mm)(1000 mm) = 1,000,000 mm².
3. Calculate the probability:
- The probability that Bob scores 100 points from his first shot is the ratio of the area where the bullet can pass through to the total area of the net.
- Probability = (Area where bullet can pass through) / (Total area of the net).
- In this case, the probability = (25π mm²) / (1,000,000 mm²).
So the probability that Bob scores 100 points from his first shot is approximately (25π) / 1,000,000.