Most related rate problems that high school students see involve three variables. The way in which things unfold usually goes as follows: One rate of change is given, the task being to find another. With three variables in the mix, the typical approach is to find a direct relationship between two of those variables and then express one in terms of the other. The third variable is then be expressed in terms of one variable only, and away we go.

There are cases, however, when all three variables must stand on their own as no direct relationship exists between any two. In such cases, two rates are provided while finding the third rate remains the goal.

The example shown below has a trough filling at a constant rate of 2 cubic feet per minute (dV/dt). The task is to determine the rate at which the depth (height) of water is changing (dh/dt) at any time. Since rates involving volume and height are directly involved here, the typical approach is to express volume in terms of height first and then differentiate. This approach (#1) will follow the image below. Since, however, I like to mix things up and provide alternative ways of thinking, a second approach will be included as well.

**The Problem:** *Determine an expression for dh/dt.*

**Approach #1**

**Approach #2**

The first approach above is more direct in this case. Having said that, awareness of approach 2 will provide leverage to deal with problems in which two independent rates are provided. In those scenarios, the substitution leading to the final expression is not carried out. I included it in the example above to show that each approach yields the same result.

Follow the link provided here to see how the rate of change in depth changes as the trough fills.

Click on the link provided here for more related rates.

Thanks for reading.

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