In this section we will discuss the only application of derivatives in this section, Related Rates. In related rates problems we are give the rate of change of one quantity in a problem and asked to determine the rate of one (or more) quantities in the problem. This is often one of the more difficult sections for students. We work quite a few problems in this section so hopefully by the end of. Sep 09,  · Problem Solving > Related Rates. Related rate problems involve equations where a relationship exists between two or more derivatives. For example, you might want to find out the rate that the distance is increasing between two airplanes. Solving related rate problems has many real life . Sep 18,  · This calculus video tutorial explains how to solve related rates problems using derivatives. It shows you how to calculate the rate of change with respect to .

## Related Rates: Example Problems Step by Step - Calculus How To

One of the hardest calculus problems that students have trouble with are related rates problems. This is because each application question has a different approach in solving the problem, and requires the application of derivatives. However once you know these 6 steps, then you should be able to solve any Calculus how to solve related rates problems in calculus rates problems you like.

Here are the following steps in solving a related rates question:. Express the asking rate look for phrases such as "how how to solve related rates problems in calculus, "at what", "find the rate", etc. Substitute the given information into the differential equation and solve for the asking rate. Question 1 : A cone is 30 cm tall, and has a radius of 5 cm. Initially it is full of water, how to solve related rates problems in calculus, but the water level falls at a constant rate of 1cm per second.

At what rate is the water draining from the cone? We are given information about the size of the cylinder, and the rate in which the water is draining. What is the rate constant of height? Well when the water is draining, the water level is going to change per second. This means the height is also going to change per second. Thus, expressing the given information and rate constant gives us:.

The reason why the rate of change of the height is negative is because water level is decreasing. Also, note that the rate of change of height is constant, so we call it a rate constant. The asking rate is basically what the question is asking for. In this case, it is asking for the rate which the water is draining from the cone. In other words, they are asking for the rate of change for the volume of the water. This leads us to write. Since we are dealing with volumes and cones, it would be wise to use the formula for the volume of a cone.

Taking the derivative of volume of a cone, we need to use the product rule and chain rule, how to solve related rates problems in calculus. First visualize the two products to be:.

However, there is a problem here. So we have to figure out a way to get rid of this. We can do this by going back to the original equation. But how does one get rid of the variable? By expressing r in terms of h. We can do this by using similar triangles. Notice that in our diagram from step 1, we can write more information about it. Notice the two triangles created from this drawing. Since the two triangles are "similar", then their ratios must be the same as well. In other words, the similar triangles formula gives:.

We just created an equation of r in terms of h using similar triangles! Now if we take this equation and substitute the r in the volume of a cone equation, then we will get:.

Now let's take a look at another calculus problem. This one deals with a sliding ladder on a wall. Question 2 : A 12 ft ladder is resting against a wall. The distance between the floor and the top of the ladder is 6 ft.

If the ladder is sliding down the wall at a rate of 2 ft per second, how to solve related rates problems in calculus, then how fast is the ladder moving away from the wall horizontally?

We are given information about the size of the ladder, and the vertical distance between the ladder and the floor. We also know the rate at which the ladder is sliding down the wall. This is important because it tells us that the distance between the ladder and the floor decreases as the ladder is down the wall. In other words. However, do we know the horizontal distance between the ladder and the wall the variable x x x? We don't but we can calculate this using Pythagoras.

Notice that:. We actually do, we just need to think about. Can a ladder physically expand or shrink? No way! So in a sense, how to solve related rates problems in calculus, the rate of the ladder increasing or decreasing is 0 because the length of the ladder is never going to change. Again, the asking rate is what the question is asking for. In this case, it's asking for the rate of change in which the ladder is moving away from the wall.

Since the picture we drew was a triangle, we probably want to use the formula of Pythagoras theorem. Using implicit differentiation and chain rule to take the derivative gives us:. Let's look at another related rates practice problems which involves a person walking away from a light pole.

This is a pretty famous related rates shadow problem! Question 3 : A how to solve related rates problems in calculus is standing near a light pole. The pole is 30 ft tall and the person is 5 feet tall. At what rate is the length of the shadowing increasing when the person is 20 ft away from the pole?

We could also identify the hypotenuse of the triangle, but that would be meaningless in this question. Many would be tempted to use the Pythagoras Theorem formula, however we see two triangles.

In fact, we could see that the two triangles are similar, so it would be wise to use similar triangles here. Note that the ratio of the legs of the big triangle and the small triangle should be the same. Even though the steps say to differentiate both sides of the equation, we first want to make the equation look nicer.

Note that:. Let's take a look at more hard calculus problems! This is one of the most famous related rates word problems that students encounter in the course. It involves a clock! How to solve related rates problems in calculus 4 : The minute hand of a clock is 10 mm long, and the hour hand is 5 mm. How fast is the distance between the hour hand and the minute hand changing at 3 pm? We are given information that the minute hand is 10 mm long and the hour hand is 5 mm, how to solve related rates problems in calculus.

No, but we can find those. If we were to divide this by 12, then that will be the angle difference between each hour, how to solve related rates problems in calculus. Note that the minute hand is at 12 and the hour hand is at 4. That means the angle between them in terms of time is 4 hours. Hence multiplying the angle we found by 4 will give us the angle of the triangle we drew above.

Well the minute hand does a full turn around the clock in 1 hour. So it must be true that. Where t t t is in minutes. Now if we were to subtract them, then we will get the difference of the rates of two angles. This gives us the rate of the angle of the triangle in the picture. However, the minute hand is catching up to the hour hand, so it is actually decreasing the angle.

So we actually have to put a negative in the rate of change of our angle. The asking rate is the rate at which the distance between the hour hand and minute hand is decreasing. In other words we are looking for:. Again, people would be tempted to use Pythagoras Theorem here. However, we are dealing with a triangle that is not always 90 degrees. Hence, we would want to use a formula that works for any triangle. The perfect formula for this would be the law of cosines. Note that the law of cosines formula for this triangle is:.

One thing to note here is that this formula gives us find d d d. Recall that:. Differentiating both sides of this law of cosines equation gives us:. We have pretty much covered the most relevant questions for calculus related rates. Of course, there are many other questions for related rates calculus. However, as long as you follow the 6 steps, you should be able to do any question you like.

If you want more practice problems, then I recommend you take a look at this link. Back to Course Index.

### How to do Calculus Related Rates? (8 Powerful Examples)

So I've got a 10 foot ladder that's leaning against a wall. But it's on very slick ground, and it starts to slide outward. And right when it's-- and right at the moment that we're looking at this ladder, the base of the ladder is 8 feet away from the base of the wall. To solve problems with Related Rates, we will need to know how to differentiate implicitly, as most problems will be formulas of one or more variables.. But this time we are going to take the derivative with respect to time, t, so this means we will multiply by a differential for the derivative of every variable! “Thank you thank you thank you!! you helped me solve my HW question!!! I’ve been pulling my hair out with trying to figure out these related rates problems and this really helped!!” ~ K. W. “Matheno's page on implicit differentiation is a life saver!” ~ J. G. “You have no .