Question

1. The table below gives the depth of water across a river measured at one metre
intervals between banks.
Distance (m) 0 1 2 3 4
Water depth (m) 0 0.5 1.6 0.9 0

Use the Trapezium rule to estimate the cross-sectional area of the river.
A river hydrologist estimates that at the place where this cross sectional data was
measured the average speed of water flow is 0.6m/s. Estimate the volume of water
which passes this section of the river in one minute.

Answer 1

The table below gives the depth of water across a river measured at one meter intervals between banks.

Distance (m) 0 1 2 3 4

Water depth (m) 0 0.5 1.6 0.9 0

Use the Trapezium rule to estimate the cross-sectional area of the river. A river hydrologist estimates that at the place where this cross sectional data was measured the average speed of water flow is 0.6m/s. Estimate the volume of water which passes this section of the river in one minute.

Solution:

We have

Because $h = h_1 - h_0 = h_2 - h_1 = h_3 - h_2 = h_4 - h_3 = 1$ then by the Trapezium rule we have

$$S = \frac {h}{2} (d _ {0} + 2 (d _ {1} + d _ {2} + d _ {3}) + d _ {4}) = \frac {1}{2} (0 + 2 \cdot (0. 5 + 1. 6 + 0. 9) + 0) = 3 (m ^ {2})$$

And finally

$$V = S \cdot v \cdot t = 3 \cdot 0. 6 \cdot 6 0 = 1 0 8 (m ^ {3})$$

Answer: 108 (m³)