Why do all related rates problems involve cones filled with water or ladders sliding down buildings?

In this video, I break one down from a Calc 1 exam solution using the Reverse Learning Technique.

## Related Rates Problem Statement

A conical paper cup 3 inches across the top and 4 inches deep is full of water. The cup springs a leak at the bottom and loses water at the rate of 2 cubic inches per minute. How fast is the water level dropping at the instant when the water is exactly 3 inches deep? Express the answer in inches per minute.

## Solution

##### From: http://www.math.utah.edu/

## Breakdown

## Summary

0:30 – **Breaking down the problem statement: **We want to know how fast is the water level dropping at the instant when the water is exactly 3 inches deep. The “how fast” is giving us an indication that it’s going to be a rate that we’re looking at.

1:04 – **Dissecting how we got the final answer**

2:30 – **Dissecting the implicit differentiation step (dV/dt)**

3:47 – **Dissecting how we got the volume equation (V) and how we used the problem geometry:** What we’re doing is relating the volume to the height using geometry (of the cone). You can use the principle of similarity between two triangles to figure out the relationship between “r” and “h”.

5:00 – **Analyzing the meaning of our two rates (dh/dt and dV/dt):** dh/dt is the speed at which the water level is changing and that’s what we’re trying to solve in this problem. dv/dt then is the speed at which the volume of the water is changing and that’s why it’s the negative 2, because the bottom is losing water at the rate of 2 cubic inches per minute.

6:14 – **Understanding the implicit differentiation step further, and the relationship to the chain rule:** We can then differentiate “V” using implicit differentiation and using the chain rule to relate dv/dt to dh/dt. Once we have that relationship, then it’s just a matter of reorganizing everything and plugging and chugging in step five.

8:05 – Understanding how we solved for “h” and eliminated “r” using similar triangle geometry

All right, so that’s what a Reverse Learning breakdown of a related rates problem looks like. Hopefully that’s given you some insight into not only how to solve a related rates problem, but also how these problems are constructed, and the underlying principles that go into solving these types of problems.

You do not need the formula for the volume of the cone to do this problem. It can be done more simply as follows: The area of the interface between the water and the air is (pi r^2), which is pi(3h/8)^2. At h=3inches, this is pi(9/8)^2. That many square inches multiplied by the rate at which the surface moves is -2cubic inches per minute. So -2cubic inches per minute divided by pi(9/8)^2square inches is the answer.