Lesson 11

Filling containers

Let’s fill containers with water.

11.1: Which One Doesn’t Belong: Solids

These are drawings of three-dimensional objects. Which one doesn’t belong? Explain your reasoning.

Four different, three-dimensional shapes labeled A, B, C, and D.  Shape "A" is a cone; Shape "B" is a sphere; Shape "C" is a cylinder; Shape "D" is a rectangular prism.

 

 

11.2: Height and Volume

Use the applet to investigate the height of water in the cylinder as a function of the water volume.

  1. Before you get started, make a prediction about the shape of the graph.

  2. Check Reset and set the radius and height of the graduated cylinder to values you choose.

  3. Let the cylinder fill with different amounts of water and record the data in the table.

     
  4. Create a graph that shows the height of the water in the cylinder as a function of the water volume.
  5. Choose a point on the graph and explain its meaning in the context of the situation.

11.3: What Is the Shape?

  1. The graph shows the height vs. volume function of an unknown container. What shape could this container have? Explain how you know and draw a possible container.
    Coordinate plane, horizontal, volume in milliliters, 0 to 100 by tens, vertical, height in centimeters, 0 to 14 by twos. Line segments from origin to 40 comma 9, then on to 100 comma 12.
  2. The graph shows the height vs. volume function of a different unknown container. What shape could this container have? Explain how you know and draw a possible container.
    Coordinate plane, horizontal, volume in milliliters, 0 to 100 by tens, vertical, height in centimeters, 0 to 14 by twos. Line segments connect origin to 10 comma 9, to 50 comma 12, to 80 comma 14.
  3. How are the two containers similar? How are they different?


The graph shows the height vs. volume function of an unknown container. What shape could this container have? Explain how you know and draw a possible container.

A graph, horizontal axis, volume in milliliters, vertical graph, height in centimeters. Graph starts at the origin, as x increases, y increases steeply before slowing down.

Summary

When filling a shape like a cylinder with water, we can see how the dimensions of the cylinder affect things like the changing height of the water. For example, let's say we have two cylinders, \(D\) and \(E\), with the same height, but \(D\) has a radius of 3 cm and \(E\) has a radius of 6 cm.
 

Two cylinders. Cylinder D, height, h, radius 3 centimeters. Cylinder E, height, h, radius, 6 centimeters.

If we pour water into both cylinders at the same rate, the height of water in \(D\) will increase faster than the height of water in \(E\) due to its smaller radius. This means that if we made graphs of the height of water as a function of the volume of water for each cylinder, we would have two lines and the slope of the line for cylinder \(D\) would be greater than the slope of the line for cylinder \(E\).

Glossary Entries

  • cylinder

    A cylinder is a three-dimensional figure like a prism, but with bases that are circles.