RELATED RATES – Cone Problem (Water Filling and Leaking)

Water is leaking out of an inverted conical tank at a rate of 10,000 \(\frac{cm^3}{min}\) at the same time water is being pumped into the tank at a constant rate. The tank has a height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 \(\frac{cm}{min}\) when the height of the water is 2 m, find the rate at which water is being pumped into the tank.

Here we have another related rates problem. This is a pretty typical problem you would see in a calculus class. There is a lot going on in this one since we have a related rates with a cone filling and leaking water. Just like I said when I discussed related rates, these problems tend to follow a specific pattern. If you need a refresher on what the four steps are just click that link to my related rates lesson. Otherwise, we’ll jump right into it.

1. Draw a sketch

You should always start a related rates problem with a drawing of the real world situation that’s being described in the problem. The problem describes an “inverted conical tank.” This just means that the tank is in the shape of an up-side-down cone. Other than that, the other facts are quite simple.

  • Water is leaking out at a rate of 10,000 \(\frac{cm^3}{min}\).
  • Liquid is being pumped into the tank at an unknown constant rate.
  • Tank’s height is 6 m.
  • Diameter of the circular opening is 4 m.
  • Water level is rising at a rate of 20 \(\frac{cm}{min}\) when the height of the water is 2 m.

So we need to put all of these facts into a drawing, which might look something like this.

conical tank related rates

The important thing to point out about our sketch is that we have two cones here.

One cone is the tank, which is the larger cone. This one will always stay the same size and is not changing. This means that its height, diameter of the base, and its volume are all constants.

The second cone is the water sitting in the bottom of the tank, which is the smaller cone. This one is changing as our liquid flows into and out of the tank. As time passes its dimensions change. Therefore, its height, diameter of the base, and volume are all functions of time.

2. Come up with your equation

Now that we have made our sketch, we need to come up with an equation that relates all of the different quantities we’re dealing with. All of the information we know about and the information we are looking for relates to a volume or measurements of some cone. The measurements are either of our water in the tank or the tank itself, but in both cases the measurements describe a cone.

What are we looking for?

The question asks us to find the rate at which water is being pumped into the tank. As it is pumped into the tank, this will impact the volume of the smaller cone which is the water sitting in the tank. Therefore, the information we are looking for will somehow relate to how quickly the volume of the liquid in the tank is changing. Or the rate of change of the volume of the small cone.

What do we know about?

We were given quite a bit of information so let’s break it into a few different pieces.

  • We know the height and base diameter of the tank. Using these, we can find its base radius and volume. Since the tank isn’t changing we know, or can easily find, any important measurement of the large cone.
  • We know how fast water is flowing out of the tank. This tells us a piece of information relating to the rate of change of the small cone. We will need to use this once we know how fast water is flowing into the tank.
  • The last information we already know is the height and the rate of change of the height of the small cone at a particular moment.

Putting it into an equation

As I mentioned above, we need to find the rate of change of the volume of the liquid in the tank. Since we know we will need to use implicit differentiation to get the rate of change, our equation needs to involve the volume of the small cone. Remember, the equation we come up with should include quantities and measurements, not rates of change yet.

So we are dealing with the volume of something that’s in the shape of a cone. Because of this we can start by looking at the formula for a volume of a cone then we’ll see if we can use this directly or if we will need to make some adjustments. The volume of a cone can be described by

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

where r is the radius of the cone’s base and h is its height.

Clearly we would be able to use this to get the rate of change of the volume since the equation already includes the volume. But this means we need to know everything else in the equation.

Other than the volume, the equation requires that we know the radius and height of the cone. We also need to know the rate of change of these other parts.

By referring back to our original drawing, you can see that we clearly already know the height and the rate of change of the height of the liquid in the tank at this specific moment. Remember, these two values are changing as liquid goes into the tank, but we know their values at this specific instant.

But what about the radius?

We don’t know the radius yet, but we can find it by comparing the small and large cones and using similar triangles. However, the problem is that we don’t know the rate of change of the radius. We could figure it out, but it would be quite complicated and there is actually something else we can do instead.

So we want to avoid using the radius, or diameter, because we don’t want to have to find its rate of change. So what else can we do?

In order to get around this, we will need to write an equation that doesn’t include the radius. But remember, it’s fine if we use the height because we know its rate of change. The important fact we need to use here is that the two cones will always form similar triangles. This will be true no matter where the water level is, as it fills up.

Consider the following drawing which represents the conical tank shown from the side.

Similar triangles

As shown in the sketch above, the top sides of these triangles are parallel. This means that each angle in the small triangle is the same as the corresponding angle in the large triangle. Therefore, these are similar triangles. Since they are similar triangles, we know one important fact:

$$\frac{x}{2}=\frac{4}{6}$$

So we can use this to solve for x.

$${x}=\frac{4}{6} \cdot 2$$

$$x=\frac{4}{3}$$

So we can see that in both of these triangles, the top side is two-thirds as long as the height of the triangle. In other words, if we multiply the height of the triangle by \(\frac{2}{3}\), this would give us the length of the top side of the triangle. Remember the top of the triangle is the diameter of the corresponding cone from our first drawing. This tells us the following relationship between the height and diameter of the cones:

$$\frac{2}{3}h=d$$

Since our goal here is to find out the relationship between the radius and the height, not the diameter and the height, we need to do one more thing. The diameter of a circle is always twice as much as the radius, so we know \(d=2r\). We can substitute this in for d and solve for r.

$$\frac{2}{3}h=2r$$

$$\frac{1}{3}h=r$$

Going back to our equation

Now we need to go all the way back to our volume equation that involved h and r.

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

Since we now have this new way to write r in terms of h, we can substitute this into the volume equation and simplify from there.

$$V=\pi \bigg( \frac{1}{3}h\bigg)^2 \frac{h}{3}$$

$$V=\pi \cdot \frac{1}{9}h^2 \frac{h}{3}$$

$$V= \frac{1}{27} \pi h^3$$

Now we have an equation involving V and h. Since we are looking for the rate of change of V, and we already know about V, h, and the rate of change of h, this equation will work perfectly.

3. Implicit differentiation

We created our equation containing all of the necessary pieces. Remember earlier I said that we are going to need to find the value of \(\frac{dV}{dt}\) in order to find how quickly liquid is going into the tank? Now is the time to do this.

As with any related rates problem, we need to take the derivative of our equation. Since we already have an equation for V, all we need to do to find \(\frac{dV}{dt}\) is take the derivative of V with respect to time. Let’s do that now.

$$\frac{d}{dt} \big[ V \big] = \frac{d}{dt} \bigg[ \frac{1}{27} \pi h^3 \bigg]$$

The left side of this equation will be quite simple, but to find the right side we need to keep a couple things in mind. We are going to take the derivative with respect to time. Therefore, we need to treat h as a function of time rather than a variable. This means that we will need to use the chain rule to take this derivative. You can see a more detailed explanation of how to do this and why we are doing it in my implicit differentiation lesson.

Also remember that \(\frac{1}{27} \pi\) is just a constant coefficient in front of our \(h^3\) term.

$$\frac{dV}{dt}= \frac{3}{27} \pi h^2 \cdot \frac{dh}{dt}$$

$$\frac{dV}{dt}= \frac{1}{9} \pi h^2 \cdot \frac{dh}{dt}$$

4. Solve for desired rate of change

The final step of a related rates problem is to solve of the rate of change the question asked for. Since \(\frac{dV}{dt}\) is already isolated we don’t need to worry about solving for the variable we are looking for.

$$\frac{dV}{dt}= \frac{1}{9} \pi h^2 \cdot \frac{dh}{dt}$$

All we need to do now is plug in the other information missing from the equation to tell us all we need to know about \(\frac{dV}{dt}\). As you can see, this mean we need to figure out the values for h and \(\frac{dh}{dt}\). Luckily, both the height and the rate the height is increasing of the small cone were both given to us.

Looking back at our drawing, which I have put below, we can see that both of these values we need are already labeled.

related rates cone

Looking at our sketch we can see that h, which is shown as the height of our small cone, is 2 m. We can also see that \(\frac{dh}{dt}\), which is shown as the speed at which the water level is rising, is 20 \(\frac{cm}{min}\). But there is one problem here. We need to be careful of our units!

Notice that the height is measured in meters and its rate of change is measured in centimeters.

We need to convert one of these two measurements so that we are either using meters or centimeters throughout the whole problem. It doesn’t matter which one we choose as long as there are the same. I will go ahead and use meters. Don’t worry, if you need the answer in centimeters I’ll discuss how to convert to that in a minute. Since \(100 \ cm = 1 \ m\) we know that

$$h=2m$$

$$\frac{dh}{dt} = 0.2 \ \frac{m}{min}$$

Putting it all together

So now we can plug these values back into our equation for \(\frac{dV}{dt}\).

$$\frac{dV}{dt}= \frac{1}{9} \pi h^2 \cdot \frac{dh}{dt}$$

$$\frac{dV}{dt}= \frac{1}{9} \pi (2)^2 \cdot 0.2$$

$$\frac{dV}{dt}= \frac{4}{9} \pi \cdot \frac{1}{5}$$

$$\frac{dV}{dt}= \frac{4 \pi}{45}$$

$$\frac{dV}{dt} \approx 0.279 \ \frac{m^3}{min}$$

But remember this isn’t quite our answer. We still have one more step. The question asked us to find the rate at which water is being pumped into the tank. We just found the rate at which the volume of water in the tank is changing.

There are three important values to think about here.

  • The rate water is being pumped into the tank.
  • Rate of change of the water actually in the tank.
  • The rate liquid is flowing out of the tank.

We are trying to find the rate water is being pumped into the tank and we already know the other two rates. It was given that water is flowing out of the tank at a rate of 10,000 \(\frac{cm^3}{min}\).

Be careful of the units

We know the rate at which the liquid is flowing out of the tank, but it uses centimeters. Since we decided earlier that we want to use meters instead, we need to convert this as well. This will be a little different though.

Think about a cube with each side length being 1 m. This cube would be 1m x 1m x 1m and would have a volume of 1 \(m^3\). Since we know that \(1 \ m = 100 \ cm\), we can also say that this same cube is 100cm x 100cm x 100cm and would actually have a volume of 1,000,000 \(cm^3\). Therefore, we can say

$$1m^3 = 1,000,000cm^3.$$

If we divide both sides of this equation by 100, we can see that

$$\frac{1}{100}m^3 = 10,000cm^3.$$

So we then can say that water is flowing out of the tank at a rate of 0.01 \(\frac{m^3}{min}\).

How it fits together

We know that \(\frac{dV}{dt} \approx 0.279 \ \frac{m^3}{min}\) is a positive number. This means that the amount of water in the tank is increasing at this instant. Since some water is flowing out of the tank at a rate of 0.01 \(\frac{m^3}{min}\), it must be flowing into the tank faster than this. If it were not, the volume in the tank would be decreasing.

In fact, you can think of the rate of change of the volume of water in the tank as the rate of water flowing in minus the rate of water flowing out.  Let’s say X represents the rate at which liquid is flowing into the tank and we can use this equation to represent the previous sentence.

$$X-0.01=0.279.$$

Solving for X tells us

$$X=0.289.$$

So water must be flowing into the tank at a rate of 0.289 \(\frac{m^3}{min}\). And this is the value the question was asking us to find!

We can also give our answer in other units

If we go back to our equation before we started plugging in anything, we can instead plug everything in terms of centimeters instead of meters. In this case we will have \(h=200cm\) and \(\frac{dh}{dt}=20 \frac{cm}{min}\).

$$\frac{dV}{dt}= \frac{1}{9} \pi h^2 \cdot \frac{dh}{dt}$$

$$\frac{dV}{dt}= \frac{1}{9} \pi (200)^2 (20)$$

$$\frac{dV}{dt} \approx 279,252.68$$

Now just like above, we need to also consider how fast the water is flowing out of the tank. This time we don’t need to convert the units though since we are using centimeters.

$$Y-10,000= 279,252.68$$

$$Y= 289,252.68$$

So we can also say that water is flowing into the tank at a rate of 289,252.68 \(\frac{cm^3}{min}\). This is more likely going to be a better answer simply because it is more precise that the previous answer we found!

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This was a good cone related rates example, but if you want some more practice you should check out my related rates lesson. At the bottom of this lesson there is a list of related rates problems and solutions. I also have several other lessons and problems on the derivatives page you can check out. If you can’t find the topic or question you’re looking for just let me know by emailing me at jakesmathlessons@gmail.com!

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