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## Why Study Calculus? – Related Rates

One of the more interesting uses of calculus is related rate problems. Problems like these demonstrate the sheer power of this branch of mathematics to answer seemingly unanswerable questions. Here we study a particular problem in related degrees and show how calculus allows us to easily arrive at a solution.

Any quantity that increases or decreases with time is a candidate for the related rate problem. It should be noted that all functions in related rate problems are time dependent. Since we are trying to find the current rate of change with respect to time, the process of differentiation (derivation) comes into play and this takes place with respect to time. Once we have outlined the problem, we can isolate the degree of change we are looking for and then solve it by differentiation. A specific example will explain this process. (Note that I took this problem from Protter/Morrey, “College Calculus,” Third Edition, and expanded on its solution and application.)

Consider the following problem: Water flows into a conical tank at a rate of 5 cubic meters per minute. The cone has a height of 20 meters and a base radius of 10 meters (the top of the cone is facing down). How fast is the water level rising when the water is 8 meters deep? Before we tackle this problem, let’s ask ourselves why we should be dealing with such a problem in the first place. Suppose a reservoir serves as part of a flow system for a dam. When a dam has an excess capacity due to flooding, such as from excessive rain or river runoff, conical reservoirs serve as outlets to release pressure on the dam walls, preventing damage to the entire dam structure.

The entire system is designed in such a way that there is an emergency procedure that is triggered when the water level in the conical tanks reaches a certain level. Certain preparation is required before performing this procedure. The workers measured the depth of the water and found it to be 8 meters deep. The question is, how long do rescuers have before the cone tanks reach capacity?

Associated degrees come into play to answer this question. If we know how fast the water level is rising at any given time, we can work out how much time we have until the reservoir will overflow. To solve this problem, let h be the depth, r the radius of the water surface, and V the volume of the water at any time t. We want to find the rate at which the height of the water is changing when h = 8. This is another way of saying that we want to know the derivative of dh/dt.

We are told that water flows at 5 cubic meters per minute. This is expressed as

dV/dt = 5. Since we are dealing with a cone, the volume of water is given by

V = (1/3)(pi)(r^2)h, so all quantities depend on time t. We see that this formula for volume depends on both the variables r and h. We want to find dh/dt which depends only on h. So we have to somehow remove the rv from the volume formula.

We can do this by drawing a picture of the situation. We see that we have a conical tank with a height of 20 meters above sea level, with a base radius of 10 meters. R can be removed by using similar triangles in the diagram. (Try drawing it to see this.) We have 10/20 = r/h, where r and h are constantly changing quantities based on the flow of water into the tank. We can solve for r to get r = 1/2h. Plugging this r value into the formula for the volume of the cone gives us V = (1/3)(pi)(.5h^2)h. (R^2 was replaced by 0.5h^2). We simplify to get

V = (1/3)(pi)(h^2/4)h or (1/12)(pi)h^3.

Since we want to know dh/dt, we take the differences to get dV = (1/4)(pi)(h^2)dh. Since we want to know these quantities with respect to time, we divide them by dt to get

(1) dV/dt = (1/4)(pi)(h^2)dh/dt.

From the original problem statement, we know that dV/dt is equal to 5. We want to find dh/dt when h = 8. So we can solve equation (1) for dh/dt by letting h = 8 and dV/dt = 5 .Entering dh/dt = (5/16pi) meters/minute or 0.099 meters/minute. Thus, the height changes at a rate of less than 1/10 of a meter every minute when the water level is 8 meters high. Emergency dam workers now have a better estimate of the current situation.

For those of you who know little about calculus, I know you will agree that problems like these demonstrate the tremendous power of the discipline. Before the bill, there would never have been a way to solve such a problem, and if it were a real impending disaster, there is no way to prevent such a tragedy. This is the power of mathematics.

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