A fountain contains a 3 m tall cylinder with holes placed at a height of 1 m and 2m. The water level is kept constant at a height of 3 m by an input pipe. As water shoots out from the tube, it strikes the fountain pool at some distance x away from the cylinder. Find x for each of the holes. If both holes are 2 cm in diameter, at what rate must the input pipe supply water.

To find the distance x at which the water strikes the fountain pool for each hole, we can use the principles of projectile motion.

1. For the first hole at a height of 1 m:
- The water will travel vertically downward for a distance of 1 m.
- The horizontal distance x can be found using the equation for horizontal motion:
x = v₀ * t, where v₀ is the initial horizontal velocity and t is the time of flight.
- To find the initial horizontal velocity v₀, we can use the conservation of energy:
Potential Energy = Kinetic Energy
m * g * 1 = 0.5 * m * v₀²
v₀ = √(2 * g * 1), where g is the acceleration due to gravity (approximately 9.8 m/s²).
- The time of flight can be calculated using the equation for vertical motion:
h = v₀y * t - 0.5 * g * t², where h is the height (1 m).
Rearranging this equation, we get:
1 = 0*t - 0.5 * 9.8 * t²
t = √(2/9.8) ≈ 0.452 seconds.
- Therefore, the horizontal distance x at which the water strikes the pool for the first hole is:
x₁ = v₀ * t = √(2 * g * 1) * 0.452.

2. For the second hole at a height of 2 m:
- The water will travel vertically downward for a distance of 2 m.
- Using the same equations as above, the time of flight for the second hole is:
t = √(2/9.8) ≈ 0.452 seconds.
- Therefore, the horizontal distance x at which the water strikes the pool for the second hole is:
x₂ = v₀ * t = √(2 * g * 1) * 0.452.

Now, to find the rate at which the input pipe must supply water, we need to consider the volume of water flowing out of each hole per second.

- The volume per second can be calculated using the formula for the volume of a cylinder:
Volume = π * r² * v, where r is the radius of the hole (1 cm = 0.01 m) and v is the velocity of the water.
- The velocity v can be found using the equation for vertical motion:
v = sqrt(2 * g * h), where g is the acceleration due to gravity and h is the height of the hole.
- Therefore, the volume per second for each hole is:
Volume₁ = π * (0.01)² * sqrt(2 * 9.8 * 1)
Volume₂ = π * (0.01)² * sqrt(2 * 9.8 * 2)
- The total rate at which the input pipe must supply water is the sum of the volumes per second for both holes:
Total volume per second = Volume₁ + Volume₂.

By plugging in the values and performing the calculations, you should be able to find the horizontal distances x₁ and x₂, as well as the rate at which the input pipe must supply water.