Here are a few more complex examples for related rates.

#### Example 1: Moving shadow

A 5ft man is walking away from a 10 ft pole at a rate of 3ft/s. How fast is the tip of his shadow moving relative to the pole when the man is 10 ft away from the pole?

##### Solution

STEP 1: Compile everything you know. Let x = distance man is from the pole, y = distance of shadow from the pole, h = height of man, and p = height of pole. We will be looking for dy/dt

STEP 2: Find the relationship between all the variables listed above. The diagram shows that this problem involves similar triangles, so…

STEP 3: Take the derivative of both sides relative to time

STEP 4: Solve for dy/dt

So when the man is 10ft from a 10ft pole and moving away from the pole at a rate of 3ft/s, the shadow is moving away from the pole at a rate of 6ft/s

#### Example 2: Distance between two objects

Bird A is moving north at a rate of 15m/s, while Bird B is moving east at a rate of 20m/s. Find the rate the distance between the two birds is changing 20 seconds later.

##### Solution

STEP 1: Make a list of everything you know. The diagram shows that we are looking for the rate of change of the hypotenuse, represented by z

STEP 2: Find a relationship between the variables listed above. Since we are looking for the rate of change of the hypotenuse…

STEP 3: Take the derivative of the relationship with respect to time

STEP 4: Find z and solve for dz/dt

When Bird A is moving north at a rate of 15ft/s and Bird B is moving east at a rate of 20m/s, the distance between the two birds is changing at a rate of 25ft/s 20 seconds later.

#### Example 3: Leaking water

Water is leaking out of a full cylindrical tank at a rate of 100cm^{3}/min. At the same time, water is being pumped into the tank at a rate of 50cm^{3}/min. If the tank has a radius of 10m and a height of 5m, what rate is the water changing when the height of the water is 1m?

##### Solution

STEP 1: Draw a diagram and organize everything you know. I = the volume of influx of water, L = the volume of water leaked, V = the volume of water left in the tank, r = radius of tank, and h = height of water in the tank.

STEP 2: Find a relationship that connects all the qualities given by the question

STEP 3: We are solving for dh/dt, so rewrite the relationship in terms of h using the ratio between the radius and height of the cylinder

STEP 4: Take the derivative of the equation relative to time

STEP 4: Solve for dh/dt

So when 50cm^{3} of water per minute is flowing out of the tank, the height of the water is decreasing at a rate of 1.33×10^^{-4}cm/min.

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