A drainage channel is being built through a market town…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A drainage channel is being built through a market town. Its cross-section is bounded below by a parabola. With the origin at the lowest point, measured across the channel and measured upwards, both in metres, the wall of the channel has the equation . During a storm the water reaches a depth of metres. The town engineer wants the area of the cross-section of the water. She decides to cut the right half of the cross-section into horizontal strips, because the equation of the wall is easy to write as in terms of . The water flows at an average speed of metres per second. The discharge, which is the volume of water passing a point each second, equals the cross-section area multiplied by the speed. The council needs these figures to decide whether the channel can carry the storm flow. Find the area of the right half, the area of the whole cross-section, and the discharge.
Need a reminder? The key idea
The worked solution
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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The strip. At height the half-width of the water is metres, and the thickness of a strip is .
Step 2. The right half. . Since , this equals square metres.
Step 3. The whole cross-section. Double it: square metres.
Step 4. The discharge. .
Check. The rectangle metres wide and metres deep has area square metres. The water fills the part above the parabola, which is square metres. This agrees.
Answer: about square metres for the half, about square metres for the whole, and cubic metres per second.
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