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Why Study Calculus? – Related Rates
One of the more interesting applications of calculus is in related rate problems. Problems like these demonstrate the full power of this branch of mathematics to answer seemingly unanswerable questions. Here we examine a particular problem in relative rates and how calculus allows us to easily solve it.
Any quantity that increases or decreases with respect to time is a candidate for the related rate problem. It should be noted that all functions in the corresponding rate problems are time dependent. Since we are trying to find the instantaneous rate of change with respect to time, the process of differentiation (taking derivatives) comes into play and it is done with respect to time. Once we’ve mapped out the problem, we can isolate the rate of change we’re looking for, and then solve it using differentiation. A specific example will clarify this process. (Please note that I have taken this problem from Proter/More, “College Calculus,” 3rd edition, and expanded on its solution and application.)
Consider the following problem: Water is flowing into a conical tank at the rate of 5 cubic meters per minute. A cone has a height of 20 m and a base radius of 10 m (the top of the cone is facing downwards). How fast does the water level rise when it is 8 meters deep? Before we tackle this problem, let’s ask why we need to address such a problem. Suppose the tank acts as part of the overflow system of the dam. When the capacity of the dam is exceeded due to flooding, say, excessive rainfall or river discharge, the conical tanks act as outlets to release the pressure on the dam walls, preventing damage to the overall dam structure.
The entire system is designed to have an emergency mechanism that kicks in when the water level in the conical tank reaches a certain level. A certain amount of preparation is required before applying this procedure. The workers measured the water depth and found it to be 8 meters deep. The question becomes how much time do emergency workers have before the conical tanks reach capacity?
To answer this question, relative rates come into play. By knowing how fast the water level is rising at any given time, we can determine how much time we have until the tank overflows. To solve this problem, we let h be the depth of the water surface, r the radius and V the volume of water at an arbitrary time. 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 getting water at the rate of 5 cubic meters per minute. It is expressed as
dV/dt = 5. We are working with a cone, given by the volume of water
V = (1/3)(pi)(r^2)h, all quantities depend on time t. We see that this volume formula depends on both the variables r and h. We want to find dh/dt, which depends only on h. Thus we need to 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 of height 20 m, with a base radius of 10 m. If we use similar triangles in the diagram we can eliminate r. (Try taking it out to see this.) We have 10/20 = r/h, where r and h represent quantities that change continuously depending on the water flow in 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 a cone, we have V = (1/3)(pi)(.5h^2)h. (We have replaced r^2 by 0.5h^2). We are easy 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 in terms of time, we divide by dt to get
(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. Thus we can solve equation (1) for dh/dt by giving h = 8 and dV/dt = 5. Inputting we get dh/dt = (5/16pi) m/. minute, or 0.099 m/min. Thus, when the water level is 8 meters above, the elevation is changing at a rate of less than 1/10 of a meter per minute. Emergency dam workers now have a better assessment of the situation at hand.
For those with some understanding of calculus, I’m sure you’ll agree that problems like these demonstrate the amazing power of this discipline. Before calculus, there was never a way to solve such a problem, and if it were a real-world impending disaster, there would be no way to avoid such a tragedy. This is the power of mathematics.
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