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But the dimensions of the cone of water must have the same proportions as those of the container. We start out by asking: What is the geometric quantity whose rate of change we know, and what is the geometric quantity whose rate of change we're being asked about? Note that the person pushing the swing is moving horizontally at a rate we know. Taking the derivative of both sides we obtain: Since the hypotenuse is constant equal to 10 , the best way to do this is to use the sine: We have seen that sometimes there are apparently more than two variables that change with time, but in reality there are just two, as the others can be expressed in terms of just two.

But sometimes there really are several variables that change with time; as long as you know the rates of change of all but one of them you can find the rate of change of the remaining one.

Exercises 6.2

As in the case when there are just two variables, take the derivative of both sides of the equation relating all of the variables, and then substitute all of the known values and solve for the unknown rate. Car A is driving north along the first road, and car B is driving east along the second road. How fast is the distance between the two cars changing?

Notice how this problem differs from example 6. In both cases we started with the Pythagorean Theorem and took derivatives on both sides. However, in example 6. In this example, on the other hand, all three sides of the right triangle are variables, even though we are interested in a specific value of each side of the triangle namely, when the sides have lengths 10 and Make sure that you understand at the start of the problem what are the variables and what are the constants.

How fast does the water level in the tank drop when the water is being drained at 3 liters per second? The foot of the ladder is pulled away from the wall at the rate of 0. How fast is the top sliding down the wall when the foot of the ladder is 5 m from the wall? How fast is the foot of the ladder approaching the wall when the foot of the ladder is 5 m from the wall? Assume that the shore is straight. At what rate is the player's distance from third base decreasing when she is half way from first to second base?

How fast is the altitude of the pile increasing when the pile is 3 cm high? The rope is being pulled through the ring at the rate of 0. How fast is the boat approaching the dock when 13 ft of rope are out? How fast is the distance between the bicyclist and the balloon increasing 2 seconds later? How fast is the shadow of a meter building shrinking at the moment when the shadow is meters long? How fast is the shadow of a 25 meter wall lengthening at the moment when the shadow is 50 meters long?

The trough is full of water. At what rate is the tip of her shadow moving? At what rate is her shadow lengthening? At what rate is the tip of his shadow moving? At what rate is his shadow shortening? The pilot uses radar to determine that an oncoming car is at a distance of exactly 1 mile from the helicopter, and that this distance is decreasing at mph. Find the speed of the car.

The pilot uses radar to determine that an oncoming car is at a distance of exactly 2 kilometers from the helicopter, and that this distance is decreasing at kph.

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A ball is falling 10 meters from the pole, casting a shadow on a building 30 meters away, as shown in figure 6. When the ball is 25 meters from the ground it is falling at 6 meters per second. How fast is its shadow moving? The copyright for this application is owned by Maplesoft.

The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft. Related Rates - Volume of Sphere. Related Rates - Volume of Sphere You can switch back to the summary page for this application by clicking here.

Learn about Maple Download Application. Maple Document Publish Date: June 20, Created In: Tell others about this application! Maple MapleSim Maple T. Terms of Use Privacy Trademarks. In this problem we eliminated the extra variable using the idea of similar triangles. This will not always be how we do this, but many of these problems do use similar triangles so make sure you can use that idea.

Related rates: Falling ladder

Note that an isosceles triangle is just a triangle in which two of the sides are the same length. In our case sides of the tank have the same length. So, we need an equation that will relate these two quantities and the volume of the tank will do it. The volume of this kind of tank is simple to compute.


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The volume is the area of the end times the depth. For our case the volume of the water in the tank is,. One for the tank itself and one formed by the water in the tank. Again, remember that with similar triangles ratios of sides must be equal.


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  • Plugging this into the volume gives a formula for the volume and only for this tank that only involved the height of the water. Also note that we converted the persons height over to 5. The tip of the shadow is defined by the rays of light just getting past the person and so we can see they form a set of similar triangles. This will be useful down the road. To do this we can again make use of the fact that the two triangles are similar to get,.

    The tip of the shadow is then moving away from the pole at a rate of 3.

    Related Rates - Volume of Sphere - Application Center

    That will happen on rare occasions. This part is actually quite simple if we have the answer from a in hand, which we do of course. Below is a copy of the sketch in the problem statement with all the relevant quantities added in. The top of the shadow will be defined by the light rays going over the head of the person and so we again get yet another set of similar triangles. Also, if the person is moving towards the wall at 2. In all the previous problems that used similar triangles we used the similar triangles to eliminate one of the variables from the equation we were working with. There is a lot to digest here with this problem.

    Be careful with the signs here. Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width.

    Determine the rate at which the radius of the balloon is increasing when the diameter of the balloon is 20 cm.

    Related Rates #1 Problem Using Implicit Differentiation

    Example 2 A 15 foot ladder is resting against the wall. How fast is the top of the ladder moving up the wall 12 seconds after we start pushing? Show Solution The first thing to do in this case is to sketch picture that shows us what is going on. Example 3 Two people are 50 feet apart. One of them starts walking north at a rate so that the angle shown in the diagram below is changing at a constant rate of 0. Show Solution This example is not as tricky as it might at first appear.