standard_description: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
problem_activity: Problem 1
A small box measures 5 in. across, 8 in. deep, and 6 in. tall. A book that measures 5
$$\times$$
8
$$\times$$
1 in. fits perfectly into the bottom of the box.
[IMG]
a.   What is the volume of one book? How many books will fit into the box? What is the volume of all of the books that will fit into the box?
b.   What is the volume of the box? How is this related to the volume of one book?
c.   A medium box measures 12 inches tall. A different book with dimensions 8
$$\times$$
10
$$\times$$
1 in. fits perfectly into the bottom of the medium box. What is the volume of the medium box?
Problem 2
The same rectangular prism is positioned in two different ways, as shown below. For each prism, use the rectangular face that the prism is “sitting on” as the base of the prism.
[IMG]
For each prism, find the area of the base and multiply by the height. Show that this is the same volume as when all three dimensions are multiplied together.
Problem 3
A rectangular prism has a base area of
$${4{3\over8}}$$
inches
$${^2}$$
, a width of
$$1 \frac{1}{4}$$
inches, and a height of
$${6 {1\over2}}$$
inches. Kirby writes the equation
$$V=4\frac{3}{8}\times1\frac{1}{4}\times6\frac{1}{2}$$
to represent the volume of the prism.
Do you agree with Kirby's equation? Explain and find the volume of the prism.


aligns: no