1.1 Changing in Tandem

Introduction

Using Graphs to Represent Relationships

Using Algebra to Add Perspective

Summary

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.

What are some different quantities that are changing in this scenario?

Sketch a labeled picture of the tank, including a snapshot of there being water in the tank prior to the tank being completely full.

What are some quantities that are changing in this scenario? What are some quantities that are not changing?

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.

$t$ | $V$

$0$ |

$2$ |

$4$ |

Finally, think about how the height, $h$, 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.

Sketch a labeled picture of the tank, including a snapshot of some water remaining in the tank prior to the tank being completely empty.

What are some quantities that are changing in this scenario? What are some quantities that are not changing?

Recall that the volume of a sphere of radius $r$ is $V = \frac{4}{3} \pi r^3$. When the tank is completely full at time $t = 0$ right before it starts being drained, how much water is present?

How long will it take for the tank to drain completely?

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. Write a sentence to explain why the data's graph appears the way that it does.

$t$ | $V$

$0$ |

$20$ |

$40$ |

$60$ |

$80$ |

$94.25$ |

Finally, think about how the height of the water changes in tandem with time. What is the height of the water when $t = 0$? What is the height when the tank is empty? 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.

Suppose we have an unusual tank whose base is a perfect sphere with radius $3$ feet, and then atop the spherical base is a cylindrical “chimney” that is a circular cylinder of radius $1$ foot and height $2$ feet, as shown in Figure. The tank is initially empty, but then a spigot is turned on that pumps water into the tank at a constant rate of $1.25$ cubic feet per minute.

Let $V$ denote the total volume of water (in cubic feet) in the tank at any time $t$ (in minutes), and $h$ the depth of the water (in feet) at time $t$.

• It is possible to use calculus to show that the total volume this tank can hold is $V_{\text{full}} = \pi(20 + \frac{38}{3}\sqrt{2}) \approx 119.11$ cubic feet. In addition, the actual height of the tank (from the bottom of the spherical base to the top of the chimney) is $h_{\text{full}} = \sqrt{8} + 5 \approx 7.83$ feet. How long does it take the tank to fill? Why? - On the blank axes provided below, sketch (by hand) possible graphs of how $V$ and $t$ change in tandem and how $h$ and $t$ change in tandem. ADD ALT TEXT TO THIS IMAGE ADD ALT TEXT TO THIS IMAGE For each graph, label any ordered pairs on the graph that you know for certain, and write at least one sentence that explains why your graphs have the shape they do. - How would your graph(s) change (if at all) if the chimney was shaped like an inverted cone instead of a cylinder? Explain and discuss.

Suppose we have a tank that is a perfect sphere with radius $6$ feet. The tank is initially empty, but then a spigot is turned on that is pumping water into the tank in a very special way: the faucet is regulated so that the depth of water in the tank is increasing at a constant rate of $0.4$ feet per minute.

Let $V$ denote the total volume of water (in cubic feet) in the tank at any time $t$ (in minutes), and $h$ the depth of the water (in feet) at given time $t$.

• How long does it take the tank to fill? What will the values of $V$ and $h$ be at the moment the tank is full? Why? - On the blank axes provided below, sketch (by hand) possible graphs of how $V$ and $t$ change in tandem and how $h$ and $t$ change in tandem. ADD ALT TEXT TO THIS IMAGE ADD ALT TEXT TO THIS IMAGE For each graph, label any ordered pairs on the graph that you know for certain, and write at least one sentence that explains why your graphs have the shape they do. - How do your responses change if the tank stays the same but instead the tank is initially full and the tank drains in such a way that the height of the water is always decreasing at a constant rate of $0.25$ feet per minute?

The relationship between the position, $s$, of a car driving on a straight road at time $t$ is given by the graph pictured at left in Figure. The car's position（You can think of the car's position like mile-markers on a highway. Saying that $s = 500$ means that the car is located $500$ feet from “marker zero” on the road.） has units measured in thousands of feet while time is measured in minutes. For instance, the point $(4,6)$ on the graph indicates that after $4$ minutes, the car has traveled $6000$ feet from its starting location.

• Write several sentences that explain the how the car is being driven and how you make these conclusions from the graph. - How far did the car travel between $t = 2$ and $t = 10$? - Does the car ever travel in reverse? Why or why not? If not, how would the graph have to look to indicate such motion? - On the blank axes in Figure, plot points or sketch a curve to describe the behavior of a car that is driven in the following way: from $t = 0$ to $t = 5$ the car travels straight down the road at a constant rate of $1000$ feet per minute. At $t = 5$, the car pulls over and parks for $2$ full minutes. Then, at $t = 7$, the car does an abrupt U-turn and returns in the opposite direction at a constant rate of $800$ feet per minute for $5$ additional minutes. As part of your work, determine (and label) the car's location at several additional points in time other than $t = 0, 5, 7, 12$. A graph of the relationship between a car's position $s$ and time $t$ A graph of the relationship between a car's position $s$ and time $t$ A graph of the relationship between a car's position $s$ and time $t$