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\author{Daniel Reinholz}

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\section*{Related Rates of Change}

It occurs often in physical applications that we know some relationship between multiple quantities, and the rate of change of one of the quantities. Our goal is to find the unknown rate of change of the other quantity. Such a situation is called a related rates problem. The key to solving related rates problems is using the known relationship between the quantities to find a relationship between their rates of change. We find the relationship between the rates of change by implictly differentiating the relationship of the quantities themselves. \\

{\bf Example 1} Supposing we are pumping up a balloon, and know that the radius of the balloon is increasing at .1 m/s. Find the rate of change of the volume of the balloon with respect to time. \\

{\bf Solution} The first step to solving this problem is identifying the quantities of interest - the radius and volume of the balloon, and their rates of change with time. Let us refer to the radius as $r$ and the volume as $V$. Since the balloon is being pumped up, the radius depends on time, so we have $r(t)$. The volume also depends on time, but indirectly, through the function $r$. Thus, we have $V(r(t))$ to describe the volume as a function of time. Our goal is to find the time rate of change of the volume $dV/dt$, and we know that $dr/dt = 0.1$. Since $V(r(t))$ is a composition of functions, to find $dV/dt$ we need to apply the chain rule, yielding 
$$\frac{dV}{dt} = \frac{dV}{dr}\frac{dr}{dt}$$
Thus, we need to find two other quantities in order to calculate $dV/dt$. We already know $dr/dt$, we so just need to find $dV/dr$. This is where we need to find a relationship between $V$ and $r$, so that we can find $dV/dr$. In this case, volume depends on radius, given by
$$V = \frac{4}{3}\pi r^3$$
Now we can find $dV/dr$ by differentiating the above equation with respect to $r$, yielding
$$\frac{dV}{dr} = 4\pi r^2$$
Multiplying the results, we find that 
$$\frac{dV}{dt} = 4\pi r^2 \cdot 0.1 = 0.4 \pi r^2$$
We see that the rate of change of the volume with respect to time depends on the current radius of the balloon. As the radius increases, the rate at which the volume increases will increase as well - the volume will begin to increase faster. Also of note is that in the above equation, we are measuring volume in units of $\text{m}^3$.\\

Although the above was a completely valid way to solve a related rates problem, we can actually solve such a problem a little more easily using the chain rule in a different way - implicit differentiation. When we setup a related rates problem this way, we find the relationship between the variables of interest, and then use implicit differentiation with respect to time, so find our result. Consider the following example. \\

{\bf Example 2} Suppose we have a vertical tank of water, with constant radius $r$. Liquid is flowing from a tap at the bottom of the tank at a rate of 3000 L/min. At what rate is the height of the water in the tank decreasing? 

{\bf Solution} First we identify that we need to consider the height $h$ of the water in the tank, and the volume $V$ of water in the tank. Since water is flowing with time, we will consider both of these functions of time, so we write $h(t)$ and $V(t)$. We know that $dV/dt = -3000$ L/min, and want to find $dh/dt$. Rather than using the chain rule, we find a relationship between the quantities $V$ and $h$, and apply implicit differentiation with respect to time. The volume of a cylindrical tank is given by
$$V = \pi r^2 h$$
assuming that the units of both sides of the equation are the same. However, $1 L = (10 \text{cm})^3$, so we a problem with our units, because on the left hand side we have $\text{cm}^3$, and on the right hand side we have $\text{m}^3$. In converting between cm and m we have the relationship
$$10 \text{ cm} = \frac{1}{10} \text{ m}$$
which implies that $1 \text{ L} = (10 \text{ cm})^3 = 1/1000 \text{ m}^3$. Or in other words, $1 L = 10^{-3} \text{ m}$. To convert the right hand side of the equation from $\text{m}^3$ to L, we need to multiply by $10^3 = 1000$. Thus, with appropriate units
$$V = 1000 \pi r^2 h$$
where volume is measured in Liters and radius and height are measured in meters. We differentiate both sides of the equation with respect to $t$, to find
$$\frac{dV}{dt} = 1000 \pi r^2 \frac{dh}{dt}$$
so that
$$\frac{dh}{dt} = \frac{dV}{dt} \frac{1}{1000 \pi r^2} = -3000 \frac{1}{1000 \pi r^2} = -\frac{3}{\pi r^2}$$
where $dh/dt$ is measured in m/min. Note that smaller the radius is, the faster the height drops. This is because with a small radius 3000 L of water leaving per minute would correspond to a volume of water with a large height, whereas when $r$ is larger the volume would have a relatively small height. \\

Having gone through two examples, we can describe the general setup of a related rates problem. 
\begin{enumerate}
\item Define our variables from the statement of a problem, and determine what variable they are dependent upon (most likely time). 
\item Gather all known information about the variables, given in the problem statement, and determine the quantity you need to find.
\item Find a relationship between the variables of interest, using some type of physical, geometric, or other type of consideration. 
\item Differentiate both sides of the relationship with respect to the independent variable (probably time), and solve for the unknown rate of change in terms of known quantities. \\
\end{enumerate}

{\bf Example 3} Assume that a hot air balloon is rising, and that the ascent of the balloon is completely vertical. The height of the balloon is being tracked by a range finder 500ft from the liftoff point. At the moment the range finder's angle is $\pi/4$, the angle is increasing at the rate of $0.14$ rad/min. How fast is the balloon rising at that instant? 

{\bf Solution} First we identify our variables of interest: the angle the range finder makes with the ground $\theta(t)$, and the height of the balloon $y(t)$. We know that $d\theta/dt |_{\theta = \pi/4} = 0.14$ rad/min, and we want to find $dy/dt |_{\theta = \pi/4}$. We can find a relationship between the quantities $\theta$ and $y$ by using the right triangle formed by the ground, the vertical path of balloon, and the range finder. This is a right triangle with a base of 500 ft, height of $y$ ft, and with angle $\theta$ adjacent to the base. Using right triangle trigonometry, we find that
$$\tan(\theta) = \frac{y}{500}$$
Differentiating both sides of the equation with respect to $t$, we find
$$\sec^2(\theta) \frac{d\theta}{dt} = \frac{1}{500}\frac{dy}{dt}$$
Thus, 
$$\frac{dy}{dt} = 500 \sec^2(\theta) \frac{d\theta}{dt}$$
It follows that
$$\frac{dy}{dt}\Big|_{\theta = \pi/4} = 500 \sec^2(\pi/4) \cdot 0.14 = 70 (\sqrt{2})^2 =140$$
where $dy/dt$ is measured in ft/min.\\

{\bf Example 4} Suppose a police cruiser is chasing a speeding car. The police cruiser is heading south approaching an intersection, and the speeding car has already passed the intersection and is heading east. When the police cruiser is 0.6 mi north of the intersection, the car is 0.8 mi to the east, and the police radar determines that the distance between them and the car is increasing at 20mph. If the cruiser is moving at 60mph when this measurement is made, what is the speed of the car?

{\bf Solution} Let $y$ denote the position of the police cruiser, and $x$ the position of car. It is convenient to define our origin at the intersection, so that the position of the cruiser will fall entirely on the vertical axis, and the position of the car on the horizontal axis. Let $s$ be the distance between the cruiser and the speeding car. We know that $dy/dt = -60$ mph, $ds/dt = 20$ mph, and we want to find $dx/dt$. In this case we have three variables of interest, and will need to forge some sort of relationship between them to find the speeding car's velocity. It turns out that we can create a right triangle, with base $x$, height $y$, and hypotenuse $s$. Using the Pythagorean theorem, it follows
$$s^2 = x^2 + y^2$$
differentiating with respect to time we find
$$2s\frac{ds}{dt} = 2x\frac{dx}{dt} + 2y\frac{dy}{dt}$$
Here we have two unknowns: $s$ and $dx/dt$. However, using the Pythagorean theorem again we find $s = \sqrt{x^2+y^2}$ so
$$\frac{dx}{dt} = \frac{1}{x}\Big(s\frac{ds}{dt} - y\frac{dy}{dt}\Big) = \frac{1}{x}\Big(\sqrt{x^2+y^2}\frac{ds}{dt} - y\frac{dy}{dt}\Big)$$
now substituting the quantities on the right hand side, we find
$$\frac{dx}{dt} = \frac{1}{0.8}\Big(\sqrt{0.8^2+0.6^2}\cdot20 - 0.6\cdot(-60)\Big) = 70$$
Thus, at the moment the cruiser measured the car's velocity, the car was going 70mph.\\

{\bf Example 5} An oil well located near the middle of a calm, shallow lake is leaking oil onto the surface of the lake at the rate of 16 cubic feet per minute. Since oil and water don't mix, as the oil leaks onto the surface of the lake it spreads out into a uniform layer, called an oil slick, that floats on top of the water. The slick is circular, centered at the leak, and 0.02 ft thick. Find the rate at which the radius of the slick is increasing when the radius is 500 and 700 ft. Why would you expect the rates of change of the radii to be different, even though the oil is being added at a constant rate?

{\bf Solution} The variables of interest are the volume $V$ and the radius $r$ of the oil slick. We know that $dV/dt = 16$, and want to find $dr/dt$. Since the surface of the oil slick is circular, and it has a height of 0.02 ft, we are dealing with a very thin, right circular cylinder. The relationship between the volume and radius of the cylinder are given by
$$V = \pi r^2 h = 0.02\pi r^2$$
Differentiating both sides of the equation with respect to $t$ we find
$$\frac{dV}{dt} = 0.04\pi r \frac{dr}{dt}$$
or that
$$\frac{dr}{dt} = \frac{1}{0.04\pi r}\frac{dV}{dt} = \frac{400}{\pi r}$$
Substituting the appropriate values for $r$ we find
$$\frac{dr}{dt}\Big|_{r=500} = \frac{400}{500 \pi} \approx 0.255 \quad \text{and} \quad \frac{dr}{dt}\Big|_{r=700} = \frac{400}{700 \pi} \approx 0.182$$
The reason why the radius is increasing at a slower rate as $r$ increases is as follows: the way in which the surface area increases is that we have concentric hollow discs being added around the initial circular surface. The larger $r$ is, the larger the volume of such a disc is, whereby the radius $r$ increases by less in order for a constant volume of oil to be added to the slick. \\

{\bf Example 6} Suppose we have two right circular cones, cone A and cone B. The height of cone A and the diameter of cone B both change at a rate of 4 cm/s, while the diameter of cone A and the height of cone B are both constant. At a particular instant, both cones have the same shape: $h = d = 10$ cm where $h$ is height and $d$ is diameter. Find the rates of change of the volume of the two cones at this time. Why would you expect the volume of the cones to be changing at different rates? 

{\bf Solution} First it is helpful to try and visualize what is happening to each of these cones. The height of cone A is increasing, so cone A is becoming taller and taller, so starts to look tall and skinny. Cone B becomes wider and wider, so starts to become short and fat. Of interest here are the height $h$, diameter $D$ (using capital $D$ as $d$ would create a notation issue), and volume $V$ for the two cones. At the particular instant of interest, we know $h_a = h_b = D_a = D_b = 10 \text{ cm}$. We also know that $dh_a/dt = 4$ cm/s and $dD_b/dt = 4$ cm/s. The formula for the volume of a right circular cone is given by
$$V = \frac{1}{3}\pi r^2 h$$
Since our known quantity is diameter, we rewrite this equation
$$V = \frac{1}{3}\pi (D/2)^2 h = \frac{\pi}{12}D^2 h$$
Now we need to be careful in this situation. In truth, we have two separate related rates problems - one for cone A and one for cone B. For cone A $D$ will be a constant, and $h$ will vary with time, while for cone B $h$ will be a constant, and $D$ will vary with time. Thus, we must differentiate this expression separately for each cone, because different quantities will depend on time. For cone A, differentiating with respect to $t$, we find the relationship
$$\frac{dV_a}{dt} = \frac{\pi}{12}D_a^2 \frac{dh_a}{dt} = \frac{\pi}{12} \cdot 10^2 \cdot 4 = \frac{100\pi}{3} \approx 104.72$$
in units of $\text{cm}^3$/s. Now for cone B, differentiating with respect to $t$, we find the relationship
$$\frac{dV_b}{dt} = \frac{\pi}{12} h_b \cdot 2 D_b \cdot \frac{dD_b}{dt} = \frac{\pi}{6} 10 \cdot 10 \cdot 4 = \frac{200 \pi}{3} \approx 209.44$$
We expect that the two cones will have different rates of change, because in one case, the volume is related to the first power of a changing quantity, and in the second it is related to the square of the changing quantity.
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