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

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 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 cm3 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 13𝜋𝑟2ℎ. You may leave 𝜋 in your answer; do not use a calculator to find a decimal answer.)

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

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 cm3/s, so 𝑑𝑉𝑑𝑡 = −15 cm3s. 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=13𝜋𝑟2ℎ

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.

Then substituting the expression for r into our relation for V:

𝑉=13𝜋(25ℎ)2ℎ=13425𝜋ℎ3=475𝜋ℎ3 3. Take the derivative with respect to time of both sides of your equation. Remember the chain rule. 𝑑𝑉𝑑𝑡=𝑑𝑑𝑡(475𝜋ℎ3)=475𝜋𝑑𝑑𝑡(ℎ3)=475𝜋(3ℎ2𝑑ℎ𝑑𝑡)=425𝜋ℎ2𝑑ℎ𝑑𝑡

Why is that dh/dt there? Open for an explanation.

Are you wondering why that 𝑑ℎ𝑑𝑡 appears? The answer is the Chain Rule.

While the derivative of ℎ3 with respect to h is 𝑑𝑑ℎℎ3 =3ℎ2, the derivative of ℎ3 with respect to time t is 𝑑𝑑𝑡ℎ3 =3ℎ2𝑑ℎ𝑑𝑡.

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)=475𝜋[ℎ(𝑡)]3

Then when we take the derivative,

𝑑𝑉(𝑡)𝑑𝑡=475𝜋𝑑𝑑𝑡[ℎ(𝑡)]3=475𝜋3[ℎ(𝑡)]2𝑑𝑑𝑡ℎ(𝑡)=425𝜋[ℎ(𝑡)]2𝑑ℎ(𝑡)𝑑𝑡

[Recall that we're looking for 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 𝑑ℎ𝑑𝑡 term.

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4. Solve for the quantity you're after.

At this point we're just substituting values. We have 𝑑𝑉𝑑𝑡 = −15 cm3s, and want to find 𝑑ℎ𝑑𝑡 at the instant when h = 10 cm.

Starting from our last expression above:

𝑑𝑉𝑑𝑡=425𝜋ℎ2𝑑ℎ𝑑𝑡𝑑ℎ𝑑𝑡=254𝜋ℎ2𝑑𝑉𝑑𝑡=254𝜋(10)2(−15)=254𝜋(100)(−15)=−1516𝜋 cm/s✓

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.

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Caution: IF you are using a web-based homework system and the question asks,

At what rate does the water level fall?

then the system may (depending entirely on how the question-writer entered their answer) already account for the negative sign, and so to be correct you probably must enter a POSITIVE VALUE: 1516𝜋 cms ✓

That's to say, if you think you did the problem correctly but the system tells you that your answer is wrong, try entering your value without the negative sign. Depending on how your instructor constructed your assignment, this convention may even vary from problem to problem in your homework set, just depending on which problems they chose, and the different answer writers for the various problems. (That can be really frustrating, we agree!) Let's be clear: 𝑑ℎ𝑑𝑡 as we found it above is a negative value, which is a key take-away here.

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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!

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