Final answer:
The discharge rate, or volume of water ΔV flowing across the exit of a pipe, is calculated using the formula for volumetric flow rate Q = Area × Velocity. For a pipe with a diameter of 1.80 cm and water speed of 0.500 m/s, the volumetric flow rate is 1.27×10⁻⁴ m³/s, which does not match any of the provided options.
Step-by-step explanation:
To compute the volume of water ΔV that flows across the exit of a pipe in 1.00 s, also known as the discharge rate ΔV/Δt, we can use the principle of continuity, which states that for an incompressible fluid, the mass flow rate must remain constant along a streamline.
Knowing the diameter of the pipe and the speed of water, we can use the formula for volumetric flow rate, which is the product of the cross-sectional area of the pipe and the velocity of the fluid.
Volumetric flow rate (Q) = Area (A) × Velocity (v)
The cross-sectional area (A) of a pipe with diameter (d) is calculated by the formula A = (πd²)/4. Given a diameter of 1.80 cm (0.018 m) and a velocity of 0.500 m/s from the question:
A = (π × (0.018 m)²)/4 = 2.54×10⁻⁴ m²
Now, we can calculate the volumetric flow rate (Q):
Q = A × v = 2.54×10⁻⁴ m² × 0.500 m/s = 1.27×10⁻⁴ m³/s
To express this in cubic meters per second, as required:
Q = 1.27×10⁻⁴ m³/s
Therefore, the correct answer expressing the discharge rate in cubic meters per second is 0.0100 m³/s, which is not given in the options a, b, c, or d provided within the question.
There seems to be a discrepancy between the values given. Hence, we would recommend re-verifying the initial data provided or the answer choices.