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## Why Study Calculus? – Related Rates

One of the most interesting applications of calculus is in problems related to rates. Problems like these demonstrate the great power of this branch of mathematics to answer seemingly unanswerable questions. Here we examine a specific problem of related rates and show how calculus allows us to find the solution quite easily.

Any amount that increases or decreases over time is a candidate for a rate-related problem. It should be noted that all functions of rate-related problems are time-dependent. Since we are trying to find an instantaneous rate of change with respect to time, the process of differentiation (taking derivatives) comes into play and this is done with respect to time. Once we map out the problem, we can isolate the rate of change we’re looking for and then solve for it by differentiation. A concrete example will make this procedure clear. (Please note that I have taken this problem from Protter/Morrey, “College Calculus,” Third Edition, and expanded 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 an altitude of 20 meters and a base radius of 10 meters (the apex of the cone is facing down). How fast is the water level rising when the water is 8 meters deep? Before we address this issue, let’s ask why we might need to address this issue. Assume that the reservoir serves as part of an overflow system for a dam. When the dam is overcapacity due to flooding as a result of, say, excessive rain or river drainage, the conical reservoirs serve as outlets to release pressure on the dam walls, preventing damage to the overall dam structure.

This whole system has been designed so that there is an emergency procedure that starts when the water levels in the conical tanks reach a certain level. Before implementing this procedure, some preparation is required. The workers have measured the depth of the water and find that it is 8 meters deep. The question is how long do emergency workers have before the cone tanks reach capacity?

To answer this question, related fees come into play. If we know how fast the water level is rising at any given time, we can determine how long 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 an arbitrary 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 dh/dt.

We are told that the water enters 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 magnitudes depend on time t. We see that this volume formula depends on the two variables ri h. We want to find dh/dt, which depends only on h. So we have to somehow remove ra from the volume formula.

We can do this by drawing a picture of the situation. We see that we have a conical tank 20 meters high, with a base radius of 10 meters. We can eliminate r if we use triangles similar to the diagram. (Try to draw it to see.) We have 10/20 = r/h, where rih represent the constantly changing quantities as a function of the water flow in the tank. We can solve for r to get r = 1/2h. If we plug this value of ra into the formula for the volume of the cone, we have V = (1/3)(pi)(.5h^2)h. (We replaced r^2 with 0.5h^2). We simplify to achieve

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

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

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

We know that dV/dt is equal to 5 from the original problem statement. We want to find dh/dt when h = 8. Thus, we can solve equation (1) for dh/dt leaving h = 8 and dV/dt = 5. Inserting we get 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 per minute when the water level is 8 meters. Emergency dam workers now have a better assessment of the situation.

For those with some understanding of calculus, I know you will agree that problems like these demonstrate the incredible power of this discipline. Before calculus, there would never have been a way to solve this problem, and if this were an impending real-world disaster, there would be no way to prevent such a tragedy. This is the power of mathematics.

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