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4 Steps to Solve Any Related Rates Problem – Part 2

In our last post, we developed four steps to solve any related rates problem.

PROBLEM SOLVING STRATEGY: Related Rates
Hide/Show Strategy
  1. Draw a picture of the physical situation.
    Don’t stare at a blank piece of paper; instead, sketch the situation for yourself. Really.
  2. Write an equation that relates the quantities of interest.
    1. Be sure to label as a variable any value that changes as the situation progresses; don’t substitute a number for it yet.
    2. To develop your equation, you will probably use:
      • a simple geometric fact (like the relation between a circle’s area and its radius, or the relation between the volume of a cone and its base-radius and height); or
      • a trigonometric function (like $\tan{\theta}$ = opposite/adjacent); or
      • the Pythagorean theorem; or
      • similar triangles.

    Most frequently (> 80% of the time) you will use the Pythagorean theorem or similar triangles.

  3. Take the derivative with respect to time of both sides of your equation. Remember the Chain Rule.
  4. Solve for the quantity you’re after.
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We introduced three examples to illustrate the basic ideas, and solved two of them there.
As promised, we’ll solve the third here.

Water Leaving a Cone Example

Here’s the problem statement, now with some additional details about the cone itself and the moment we’re interested in:

Water leaks out of a cone at 15 cubic-cm each second.Water in a Cone Example. Given: An inverted cone is 20 cm tall, has an opening radius of 8 cm, and was initially full of water. It is now being drained of water at the constant rate of 15 cm$^3$ each second. The water’s surface level falls as a result. Question: At what rate is the water level falling when the water is halfway down the cone? (Note: The volume of a cone is $\dfrac{1}{3}\pi r^{2}h$. You may leave $\pi$ in your answer; do not use a calculator to find a decimal answer.)

Let’s use our Problem Solving Strategy to answer the question.

The cone, with sizes labeled.
1. Draw a picture of the physical situation.
See the figure.

When a quantity is decreasing, we have to make the rate negative.
We are given that the volume of water in the cup is decreasing at the rate of 15 cm$^3$/s, so $\dfrac{dV}{dt} = -15\, \tfrac{\text{cm}^3}{\text{s}}$. Remember that we have to insert that negative sign “by hand” since the water’s volume is decreasing.


2. Write an equation that relates the quantities of interest.
A. Be sure to label as a variable any value that changes as the situation progresses; don’t substitute a number for it yet.
The height of the water changes as time passes, so we’re going to keep that height as a variable, h.

B. To develop your equation, you will probably use . . . similar triangles.
We have a relation between the volume of water in the cup at any moment, and the water’s current height, h:

$$V = \frac{1}{3} \pi r^2h $$

Notice that this relation expresses the water’s volume as the function of two variables, r and h. We can only take the derivative with respect to one variable, so we need to eliminate one of those two. Since the question asks us to find the rate at which the water is falling when its at a particular height, let’s keep h and eliminate r as a variable using similar triangles.


Begin subproblem to eliminate r as a variable.
The water's volume and the full cone form similar triangles.
The figure is the same as in Step 1, but with the rest of the cone removed for clarity. Note that there are two triangles, a small one inside a larger one. Because these are similar triangles, the ratio of the base of the small triangle to that of the big triangle $\left(\dfrac{r}{8} \right)$ must equal the ratio of the height of the small triangle to that of the big triangle $\left(\dfrac{h}{20} \right) $:
\begin{align*}
\frac{r}{8} &= \frac{h}{20} \\[8px] r &= \frac{8}{20} h \\[8px] &= \frac{2}{5} h
\end{align*}
End subproblem.


Then substituting the expression for r into our relation for V:
\begin{align*}
V &= \frac{1}{3} \pi \left(\frac{2}{5} h \right)^2h \\ \\
&= \frac{1}{3}\frac{4}{25} \pi h^3 \\ \\
&= \frac{4}{75} \pi h^3
\end{align*}
3. Take the derivative with respect to time of both sides of your equation. Remember the chain rule.
\begin{align*}
\frac{dV}{dt} &= \frac{d}{dt}\left(\frac{4}{75} \pi h^3 \right) \\ \\
&= \frac{4}{75} \pi \frac{d}{dt}\left(h^3 \right) \\ \\
&= \frac{4}{75} \pi \left(3h^2 \frac{dh}{dt} \right) \\ \\
&= \frac{4}{25} \pi h^2 \frac{dh}{dt}
\end{align*}

Why is that dh/dt there? Open for an explanation.
Are you wondering why that  $\dfrac{dh}{dt}$  appears? The answer is the Chain Rule.

While the derivative of $h^3$ with respect to h is  $\dfrac{d}{dh}h^3 = 3h^2$, the derivative of $h^3$ with respect to time t is  $\dfrac{d}{dt}h^3 = 3h^2\dfrac{dh}{dt}$.

Remember that h is a function of time t: the water’s height decreases as time passes. We could have captured this time-dependence explicitly by always writing the water’s height as h(t), and then explicitly showing the water’s volume in the cone as a function of time as
$$V(t) = \frac{4}{75} \pi [h(t)]^3$$
Then when we take the derivative,
\begin{align*}
\frac{dV(t)}{dt} &= \frac{4}{75} \pi \frac{d}{dt}[h(t)]^3 \\ \\
&= \frac{4}{75}\pi\,3[h(t)]^2 \frac{d}{dt}h(t)\\ \\
&= \frac{4}{25}\pi [h(t)]^2 \frac{dh(t)}{dt}
\end{align*}

[Recall that we’re looking for  $\dfrac{dh(t)}{dt}$  in this problem.]

Most people find that writing the explicit time-dependence V(t) and h(t) annoying, and so just write V and h instead. Regardless, you must remember that h depends on t, and so when you take the derivative with respect to time the Chain Rule applies and you have the  $\dfrac{dh}{dt}$ term.

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4. Solve for the quantity you’re after.
At this point we’re just substituting values. We have  $\dfrac{dV}{dt} = -15 \, \tfrac{\text{cm}^3}{\text{s}}$, and want to find  $\dfrac{dh}{dt}$ at the instant when h = 10 cm.
Starting from our last expression above:

\begin{align*}
\frac{dV}{dt} &= \frac{4}{25} \pi h^2 \frac{dh}{dt} \\ \\
\frac{dh}{dt} &= \frac{25}{4\pi h^2} \frac{dV}{dt} \\ \\
&= \frac{25}{4\pi (10)^2} (-15) \\ \\
&= \frac{25}{4\pi (100)} (-15) \\ \\
&= -\frac{15}{16\pi} \text{ cm/s} \quad \cmark
\end{align*}
The negative value indicates that the water’s height h is decreasing, which is correct.

Notice how our “Four Steps to Solve Any Related Rates Problem” led us straightforwardly to the solution. This is the strategy we use time and again; you can too.

Time to practice

You need to practice for yourself, pencil in your hand, before your exam.

Of course just reading our solution, or watching someone else solve problems, won’t really help you get better at solving calculus problems. Instead you need to practice for yourself, pencil in your hand, so you can get stuck and make mistakes and do all the other things people do when they’re learning something new. (And ideally do all those things before you’re taking an exam!) We have lots of problems for you to use, each with a complete step-by-step solution.

For more example problems with complete solutions, please visit our free Related Rates page!


We’d love your comments:

  • What tips do you have to share about solving Related Rates problems?
  • Or what questions do you have?
  • Or how can we make posts such as this one more useful to you?

Please comment below!


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Jake
8 months ago

Just saved my life for a calc midterm

Matheno
Editor
Reply to  Jake
8 months ago

Thanks, Jake!! We’re happy to have helped, and hope your midterm goes well!

Meola
1 year ago

I’ve been trying to learn this for days now and I’ve finally got it! Thank you!!

Matheno
Editor
Reply to  Meola
1 year ago

You’re very welcome, Meola. We’re happy to have helped! : )

Jason Kirsch
3 years ago

I have a similar problem, but I can’t seem to get the correct answer. My cone has a height of 8 inches and a diameter of 12inches. If the liquid pours out at 4 cubic inches a second, how fast is the level of the liquid dropping when the liquid is 3 inches deep?

Barrett
4 years ago

how are you eliminating r as a variable using r/h? where does that formula come from?
just trying to figure out how r and h are related in that step, as it is not listed where that part comes from.

Evangelista Metropolitansky
5 years ago

Cool, do you have a page for optimization problems as well? Thanks!