Problem of the Day
In Life is Strange, Max Caulfield discovered her ability to rewind time at the beginning of the series. Such ability allowed her to tamper with time, thus saving her friend, Chloe Price, from being killed at that present moment. Yet unbeknownst to her, a series of catastrophic events ensued down the line, culminating in the inevitable tornado at her home town, Arcadia Bay. (This isn’t really related to the problem, I just liked to write it down anyway.)
How would you go above solving this problem?
Part A
We want to represent the volume of water in the trough in terms of θ. Given the length of 8m of the water trough, we are interested in finding an expression for the shaded area as shown in the figure above.
And multiplying it by 8 should give us the desired expression. Recall in calculus that we deal with the angles in radians. To find an expression for the shaded area, consider it as the difference between the sector and the triangle.
The above can be done by using the formulas for finding the sector and the triangle of that sector. The radius is 0.5.
As you can see, upon multiplying the whole expression by 8, we should arrive at the following.
This step is crucial as it builds the foundation for finding the rate of change in Part B.
Part B
The question would like us to compute the rate at which the water level rises when the water is 25cm deep.
Notice the first piece of information given at the start of the question. Water is being pumped into the trough at a constant rate of 0.1m^3 per minute.
Our goal is to find the rate of change in the height of the water level.
Let’s use the hint from the question and find the rate of change of the angle. Using the chain rule, we should get the following expression.
Do you see how Part A helps us? We can find dθ/dV by first computing dV/dθ and taking its reciprocal.
And therefore, we get this.
How is this going to help us find dh/dt? Again we can apply the chain rule to dh/dt.
We have already found dθ/dt. All we have to do now is to find a relation between h and θ and subsequently find dh/dθ.
Let’s take a look at the triangle in the middle.
The radius is 0.5 and the height of the triangle shaded in orange is 0.5 - h.
From this triangle, we can establish the following relation.
So what is dh/dθ? Using basic differentiation rules, we should get this.
Now that we have all these expressions in terms of θ, let’s put them together.
Finally, our job is to use the information h = 0.25.
With this in mind, we apply trigonometric formulae to find the expressions sin(θ/2) and cosθ.
Now that we have all the numbers, let’s plug them in!
And we are done. Good job.
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Happy reading,
Barry 🍩