To set up a related rates problem using a real-life example, follow these simple steps:

1. Identify the Variables

First, figure out what things are changing over time.

Some common things to look at are:

• distance
• volume
• height
• radius

For example, if you're dealing with a water tank, you might want to consider the height of the water and how much water is in the tank.

2. Establish Relationships

Next, find how these changing things are connected.

This usually involves using some basic formulas.

For a cylindrical water tank, the volume ($V$) of water can be found with this formula:

$$V = \pi r^2 h$$

Here, $r$ stands for the radius, and $h$ stands for the height.

3. Differentiate with Respect to Time

Now, we need to take a look at how fast these things are changing over time ($t$).

We can use something called implicit differentiation along with the chain rule to connect the rates of change.

If both the height and radius of the water tank are changing, we can differentiate like this:

$$\frac{dV}{dt} = \pi \left(2rh \frac{dr}{dt} + r^2 \frac{dh}{dt}\right)$$

4. Substitute Known Values

Then, plug in the values you already know.

For example, if you know that $r = 3$ cm and $\frac{dh}{dt} = 2$ cm/min, you can replace those values in the equation.

5. Solve for the Unknown Rate

After plugging in the known values, figure out the unknown rate.

In this case, you might be looking to find out $\frac{dV}{dt}$, which tells you how fast the volume is changing.

Conclusion

By learning how to find relationships, differentiate, and substitute values, you can effectively solve related rates problems in calculus.