A river is 6 metres wide and is flowing at a uniform velocity of 2𝑚/𝑠. The depth of a cross section of the river, measured at 1𝑚 intervals, is given in metres as:0 ，1.5 ，2 ，2.8 ，2.8 ，1.9， 0. Use the trapezium rule to approximate the rate of flow of the river, that is, the rate of change of volume。
Question
A river is 6 metres wide and is flowing at a uniform velocity of 2𝑚/𝑠. The depth of a cross section of the river, measured at 1𝑚 intervals, is given in metres as:0 ，1.5 ，2 ，2.8 ，2.8 ，1.9， 0. Use the trapezium rule to approximate the rate of flow of the river, that is, the rate of change of volume。
Solution
The trapezium rule is a method used to approximate the definite integral of a function. It works by estimating the region under the graph of the function as a trapezium and calculating its area.
Here are the steps to solve the problem:
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First, we need to calculate the area of the cross-section of the river using the trapezium rule. The formula for the trapezium rule is:
Area ≈ (h/2) * (y0 + 2y1 + 2y2 + ... + 2yn-1 + yn)
where h is the width of each interval, y0 and yn are the first and last values, and y1 to yn-1 are all the other values.
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In this case, h = 1m (the interval at which depth is measured), y0 = 0m, y1 = 1.5m, y2 = 2m, y3 = 2.8m, y4 = 2.8m, y5 = 1.9m, and yn = 0m.
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Substituting these values into the formula gives:
Area ≈ (1/2) * (0 + 21.5 + 22 + 22.8 + 22.8 + 2*1.9 + 0) = 11.6 square meters
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The rate of flow (or volume change) of the river is the cross-sectional area multiplied by the velocity of the river. In this case, the velocity is given as 2m/s.
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Therefore, the rate of flow ≈ Area * velocity = 11.6 m² * 2 m/s = 23.2 cubic meters per second.
So, the rate of flow of the river is approximately 23.2 cubic meters per second.
Similar Questions
A river is 6 metres wide and is flowing at a uniform velocity of 2𝑚/𝑠. The depth of a cross section of the river, measured at 1𝑚 intervals, is given in metres as:0 ，1.5 ，2 ，2.8 ，2.8 ，1.9， 0. Use Simpson’s rule to approximate the rate of flow of the river, that is, the rate of change of volume。
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