# 11.9 and 11.10: Volume

Read section 11.9, on Principles for Determining Volume. The book discusses how the same principles of area: moving, additivity, and Cavalieri's principle of shearing, still hold true when studying volume.

1. If you move a solid shape rigidly without stretching or shrinking it, then its volume does not change.

2. If you combine a finite number of solid shapes without overlapping them, then the volume of the resulting solid shape is the sum of the volumes of the individual solid shapes.

Cavalieri's Principle: When you shear a solid shape (sliding infinitesimally small slices) the volume of the original and sheared shapes are equal.

Learnzillion is a great resource for videos and activities. They even have lesson plans, all aligned to common core.

Activity 1: Go to learnzillion.com, and type in DJC9N5R into the search box.

Activity 2: Go to learnzillion.com, and type in T3VFHMQ into the search box.

After you read through these lesson plans, answer the question in the form below, starting with **11.9-** in your response. In what other real world situations might you need to add layers of varying dimensions together to find the total volume?

**Archimedes Principle** says that an object that floats displaces an amount of water that weighs as much as the object. We can determine the volume of an object by submersing the object in a known level of water and measure how much the water level goes up.

# 11.10

Prisms and Cylinders use the formula Volume=height x area of the base.

Pyramids and cones join a base with a point. The height of the object is the perpendicular distance between the point and the plane containing the base. (It is important not to use a slant height when figuring out volume). The formula is Volume= (1/3) x height x area of the base.

I use this activity with my class when we are first going over Volume of Pyramids. If you have access to a printer, you should print out the pattern and try it. Otherwise, we can take a look when I get back.

Read over section 11.10, pages 632-634.

Think about it: How many 2x2x2 cm cubes can be stacked neatly in an 8x10x12 cm box? Will there be any empty space?

Try out this problem:

Suppose that a tube of toothpaste contains 15 cubic centimeters of toothpaste and that the circular opening where the toothpaste comes out has a diameter of 5/16 inch. If every time you use brush your teeth you squeeze out a .5 inch long piece of toothpaste, how many times can you brush your teeth with this tube?

Hint: What is the volume of toothpaste you use each time your brush your teeth? *You don't need to submit this one.*

(See the answer in the book, page 634 number 3)

Complete problem Number 3 on page 631, and Number 6 on page 636.

PG 631 Number 3: A fish tank in the shape of a rectangular prism is 40cm wide and 0 cm tall. At first, the tank is empty. Then, some stones with a total volume of 15,000cm^3 are put in the tank. Finally, 120 liters of water are poured into the tank. At that point, the tank is 3/4 full. How wide is the tank? Explain your reasoning.

Pg 636 Number 6: A cake recipe will make a round cake that is 6 inches in diameter and two inches high.

a)If you use the same recipe, but pour the batter into a round cake pan that is 8 inches in diameter, how tall will the cake be?

b) Suppose you want to use the same cake recipe to make a rectangular cake. If you use a rectangular pan that is 8 inches wide and 10 inches long, and if you want the cake to be about 2 inches tall, then how much of the recipe should you make? (For example, should you make twice as much of the recipe, half as much, three-quarters as much, or some other amount?) Give an approximate, but practical answer.

In addition: Pick one other practice problem from section 11.10 that you think is a good assessment of some of the skills we have been learning. Complete a solution write-up, posting your thoughts about the problem on the Blackboard Thread. Respond thoughtfully to at least two other people's posts.