We explore how mathematics, and graphs in particular, allow us to describe and investigate the relationships between two variables changing in tandem.
Mathematics is the art of making sense of patterns. One way that patterns arise is when two quantities are changing in tandem. In this setting, we may make sense of the situation by expressing the relationship between the changing quantities through words, through images, through data, or through a formula.
In the previous activity, we saw how several changing quantities were related in the setting of an aquarium filling with water: time, the depth of the water, and the total amount of water in the tank are all changing, and any pair of these quantities changes in related ways. One way that we can make sense of the situation is to record some data in a table. For instance, observing that the tank is filling at a rate of cubic feet per minute, this tells us that after minute there will be cubic feet of water in the tank, and after minutes there will be cubic foot of water in the tank, and so on. If we let denote the time in minutes and the amount of water in the tank at time , we can represent the relationship between these quantities through a table.
We can also represent this data in a graph by plotting ordered pairs on a system of coordinate axes, where represents the horizontal distance of the point from the origin, , and represents the vertical distance from . The visual representation of the table of values is seen in the graph below.
Sometimes it is possible to use variables and one or more equations to connect quantities that are changing in tandem. In the aquarium example from the preview activity, we can observe that the volume, , of a rectangular box that has length , width , and height is given by
and thus, since the water in the tank will always have length feet and width feet, the volume of water in the tank is directly related to the depth of water in the tank by the equation Depending on which variable we solve for, we can either see how depends on through the equation , or how depends on via the equation . From either perspective, we observe that as depth or volume increases, so must volume or depth correspondingly increase.
One of the ways that we make sense of mathematical ideas is to view them from multiple perspectives. Sometimes we use different means to establish a point of view: words, numerical data, graphs, or symbols. In addition, sometimes by changing our perspective within a particular approach we gain deeper insight.
Here is a table of data obtained algebraically of the height of the water in the cone at different times. For more information about how this table was derived, see https://activecalculus.org/prelude/sec-_changing-_in-_tandem.html
Table How time, volume, and height change in a conical tank
Plotting this data on two different sets of axes allows us to see the different ways that and change with . First we graph how volume changes over time.
Volume increases at a constant rate, as seen by the straight line appearance of the points in the graph above.
Now let’s graph how height changes over time.
We observe that the water’s height increases in a way that it rises more slowly as time goes on, as shown by the way the curve the points lie on in the graph below “bends down” as time passes.
These different behaviors make sense because of the shape of the tank. Since at first there is less volume relative to depth near the cone’s point, as water flows in at a constant rate, the water’s height will rise quickly. But as time goes on and more water is added at the same rate, there is more space for the water to fill in order to make the water level rise, and thus the water’s heigh rises more and more slowly as time passes.
Consider a tank in the shape of a sphere where the tank’s radius is feet. Suppose that the tank is initially completely full and that it is being drained by a pump at a constant rate of cubic feet per minute.
