Active Prelude to Calculus

Chapter 1 – Relating Changing Quantities

1.1 Changing in Tandem

Motivating Questions

• If we have two quantities that are changing in tandem, how can we connect the quantities and understand how change in one affects the other?
• When the amount of water in a tank is changing, what behaviors can we observe?

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.

Preview Activity 1.1.1. Suppose that a rectangular aquarium is being filled with water. The tank is 4 feet long by 2 feet wide by 3 feet high, and the hose that is filling the tank is delivering water at a rate of 0.5 cubic feet per minute.

a. What are some different quantities that are changing in this scenario?
b. After 1 minute has elapsed, how much water is in the tank? At this moment, how deep is the water?
c. How much water is in the tank and how deep is the water after 2 minutes? After 3 minutes?
d. How long will it take for the tank to be completely full? Why?

1.1.1 Using Graphs to Represent Relationships 

In Preview Activity 1.1.1, 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 0.5 cubic feet per minute, this tells us that after 1 minute there will be 0.5 cubic feet of water in the tank, and after 2 minutes there will be 1 cubic foot of water in the tank, and so on. If we let t denote the time in minutes and V the amount of water in the tank at time t, we can represent the relationship between these quantities through Table 1.1.3.

We can also represent this data in a graph by plotting ordered pairs (t ,V) on a system of coordinate axes, where t represents the horizontal distance of the point from the origin, (0, 0), and V represents the vertical distance from (0, 0). The visual representation of the table of values from Table 1.1.3 is seen in the graph in Figure 1.1.4. 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, V , of a rectangular box that has length l, width w, and height h is given by

V  = l · w · h,

and thus, since the water in the tank will always have length l  4 feet and width w  2 feet, the volume of water in the tank is directly related to the depth of water in the tank by the equation

V  4 · 2 · h  8h.

Depending on which variable we solve for, we can either see how V depends on h through the equation V  8h, or how h depends on V via the equation h 1 8 V . From either perspective, we observe that as depth or volume increases, so must volume or depth correspondingly increase.

Activity 1.1.2. Consider a tank in the shape of an inverted circular cone (point down) where the tank’s radius is 2 feet and its depth is 4 feet. Suppose that the tank is being filled with water that is entering at a constant rate of 0.75 cubic feet per minute.

a. Sketch a labeled picture of the tank, including a snapshot of there being water in the tank prior to the tank being completely full.
b. What are some quantities that are changing in this scenario? What are some quantities that are not changing?
c. Fill in the following table of values to determine how much water, V , is in the tank at a given time in minutes, t, and thus generate a graph of the relationship between volume and time by plotting the data on the provided axes.

d. Finally, think about how the height of the water changes in tandem with time. Without attempting to determine specific values of h at particular values of t, how would you expect the data for the relationship between h and t to appear? Use the provided axes to sketch at least two possibilities; write at least one sentence to explain how you think the graph should appear.