I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates. I taught my students to answer questions such as the following (taken, more or less, from the textbook):
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A man starts walking due north at 5 ft/sec from a Point A. Ten seconds later, a woman starts walking south at 4 ft/sec from a point 20 ft due east of Point A. How fast are they moving apart when the woman has been walking for ten seconds?
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A 6' man walks away from a 20' lamppost at a speed of 5 ft/sec. How fast is the distance between the tip of his shadow and the top of the post changing when he is 40' away?
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A baseball player runs from first base to second at 20 ft/sec, and simultaneously another baseball player runs from third base to home at the same speed. How fast are they approaching each other after one second?
To put my question bluntly: Who cares?
My students do, but only because they know these questions will appear on their exams. The baseball question (or something very similar) is actually an exercise in Stewart, and I struggled in vain to imagine a situation in which the manager of a baseball team would need to know the answer.
This is in stark contrast to many other topics addressed in first-year calculus — optimization, basic differential equations, etc. — which are realistic models of questions of natural interest in business, biology, etc. Basically, all the related rates questions seemed to be cooked up in response to the fact that calculus students now knew a method to solve them.
My question is in the title. Can anyone share any related rates questions which don't seem quite as contrived, and which might naturally seem interesting and motivated to a typical class of college freshmen?
Thank you!
Best Answer
The skills that students are practicing in related rates problems are:
Differentiating a known equation implicitly with respect to time.
Interpreting the time derivative of a quantity as a rate of change.
The main reason that related rates problems feel so contrived is that calculus books do not want to assume that the students are familiar with any of the equations of science or economics. Every related rates problem inherently involves differentiating a known equation, and the only equations that the calculus book assumes are the equations of geometry.
Thus, you can find related rates problems involving various area and volume formulas, related rates problems involving the Pythagorean Theorem or similar triangles, related rates problems involving triangle trigonometry, and so forth. A few of these problems are compelling -- for example, computing the speed of an airplane based on ground observations of its altitude and apparent angular velocity -- but most of them do feel a bit contrived.
The reality, of course, is that students are familiar with many of the basic equations and concepts of science and economics, and there's no rule against using these in problems. For example, you can make up all sorts of compelling related rates problems by starting with any physics or chemistry equation and imagining a situation where you might want to take its derivative:
The kinetic energy of an object is $K = \frac{1}{2}mv^2$. If the object is accelerating at a rate of $9.8 \text{m}/\text{s}^2$, how fast is the kinetic energy increasing when the speed is $30 \;\text{m}/\text{s}$?
An ideal gas satisfies $PV = nRT$, where $n$ is the number of moles and $R \approx 8.314\;\; \text{J}\; \text{mol}^{-1} \text{K}^{-1}$. Give the rate at which the temperature and volume of the gas are increasing, and then ask about the rate of change in pressure when the volume and temperature reach certain amounts.
The total energy stored in a capacitor is $\frac{1}{2} Q^2 / C$, where $Q$ is the amount of charge stored in the capacitor and $C$ is the capacitance. Give the value of $C$ and the rate at which $Q$ is decreasing, and ask about the rate at which the capacitor is losing energy when the energy is a certain amount.
In astronomy, the absolute magnitude $M$ of a star is related to its luminosity $L$ by the formula $$ M \;=\; M_{\text{sun}} -\; 2.5\; \log_{10}(L/L_{\text{sun}}). $$ where $M_{\text{sun}} = 4.75$ and $L_{\text{sun}} = 3.839 \times 10^{26} \text{watts}$. (Note that, by convention, brighter stars have lower magnitude.) If the absolute magnitude of a variable star is decreasing at a rate of $0.09 / \text{week}$, how quickly is the luminosity of the star increasing when the magnitude is $3.8$?
It's easy to make these up: just think of any equation in science or economics whose derivative might be interesting. Wikipedia and/or science textbooks can be helpful for finding equations from a wide variety of fields.
Edit: I have compiled a list of these problems in the form that I use them in my classes, and posted them on my professional web page.