Why study calculus? – Related Rates

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

Any quantity that increases or decreases with respect to time is a candidate for a related rate problem. It should be noted that all the functions in the related rate problems are time dependent. Since we are trying to find an instantaneous rate of change with respect to time, the differentiation process (taking derivatives) comes into play and this is done with respect to time. Once we map the problem, we can isolate the rate of change we’re looking for and then solve using differentiation. A specific example will clarify this procedure. (Note that I have taken this problem from Protter/Morrey, “College Calculus”, 3rd edition, and expanded on the solution and application of it.)

Consider the following problem: Water flows into a conical tank at the rate of 5 cubic meters per minute. The cone has a height of 20 meters and a base radius of 10 meters (the apex of the cone faces downwards). How fast is the water level rising when the water is 8 meters deep? Before solving this problem, let’s ask ourselves why we might need to address such a problem. Well, suppose the tank serves as part of a dam overflow system. When the dam has excess capacity due to flooding as a result of, say, excessive rainfall or river drainage, the conical tanks serve as outlets to release pressure on the dam walls, preventing damage to the overall structure of the dam. .

This entire system has been designed so that there is an emergency procedure that is activated when the water levels in the conical tanks reach a certain level. Before implementing this procedure, a certain amount of preparation is necessary. The workers have taken a measurement of the depth of the water and find that it is 8 meters deep. The question is how long do emergency workers have before the conical tanks reach capacity?

To answer this question, related fees come into play. By knowing how fast the water level is rising at any moment, we can determine how long we have until the tank overflows. 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.

They give us that the 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 the water is given by

V = (1/3)(pi)(r^2)h, such that all quantities depend on time t. We see that this volume formula depends on both variables r and h. We want to find dh/dt, which only depends on h. Therefore, we need to somehow eliminate r in 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 if we use similar triangles in the diagram. (Try to draw this to see this). We have 10/20 = r/h, where r and h represent the constantly changing quantities based on the flow of water into the tank. We can solve for r to get r = 1/2h. If we plug this value of r into the formula for the volume of the cone, we have V = (1/3)(pi)(.5h^2)h. (We have replaced r^2 with 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 differentials to get dV = (1/4)(pi)(h^2)dh. Since we want to know these quantities 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 statement of the problem. We want to find dh/dt when h = 8. Therefore, we can solve equation (1) for dh/dt by letting h = 8 and dV/dt = 5. Inputting we get dh/dt = (5/16pi)meters/ minute, or 0.099 meters/minute. Therefore, 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 workers now have a better assessment of the situation at hand.

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