Water flows into a large tank with a flat bottom at the rate of 10 m³/s. Water is also leaking out of a hole of area 1 cm² at its bottom. If the height of the water in the tank remains steady, then this height is?

Since the height of the water column is constant, therefore, water inflow rate (Qin) = water outflow rate
Qin = 10 m³/s
Qout = Au = 10⁻⁴ m² × √(2gh) (where A is the area of the hole, u is the velocity of water leaking out, g is acceleration due to gravity, and h is the height of water)
10 = 10⁻⁴ × √(2gh)
10⁵ = √(2gh)
10¹⁰ = 2gh
h = 10¹⁰ / (2g)
Assuming g ≈ 10 m/s², we have:
h = 10¹⁰ / (2 × 10) = 5 × 10⁹ m
This result is unrealistic. Let's re-examine the units. The inflow rate is given in m³/s, and the outflow area is in cm². Let's convert the area to m²:
Area (A) = 1 cm² = 1 × 10⁻⁴ m²
Now, let's use the equation for the outflow rate:
Qout = A√(2gh)
Since Qin = Qout:
10 m³/s = (1 × 10⁻⁴ m²)√(2gh)
10⁵ = √(2gh)
10¹⁰ = 2gh
Assuming g = 9.81 m/s²:
h = 10¹⁰ / (2 * 9.81) ≈ 5.097 × 10⁸ m
This is still unrealistic. There must be a mistake in the problem statement or given data. Let's assume the inflow rate is 10 L/s = 0.01 m³/s.
0.01 = 10⁻⁴√(2gh)
100 = √(2gh)
10000 = 2gh
h = 10000 / (2 * 9.81) ≈ 509.7 m
This is also unrealistic. There's an error in the original solution and likely in the problem statement or units.
Let's assume the inflow rate is 10 liters/second (0.01 m³/s):
0.01 m³/s = (10⁻⁴ m²)√(2gh)
100 = √(2gh)
10000 = 2gh
Assuming g = 9.81 m/s²:
h = 10000 / (2 * 9.81) ≈ 509.68 m
This height is still implausibly large. There is an error in the problem statement or data provided. The given options (2.9 cm, 1.7 cm, 5.1 cm, 4 cm) seem to indicate a much smaller inflow rate than 10 m³/s. Without corrected data, a precise solution is impossible.