Volume Formula for a Cylinder
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- John Mosley
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1 ACTIVITY 1.1 Volume Formula for a Analyze the prisms shown. Triangular Prism Rectangular Prism Pentagonal Prism Hexagonal Prism 1. What pattern do you see as the number of sides of the base increases? Prisms and cylinders both have two bases and a constant height between the bases. 2. Because cylinders and prisms are similar in composition, their volumes are calculated in similar ways. a. Write the formula for the volume of any right prism. Define all variables used in the formula. b. Make a conjecture about how you will calculate the volume of a right cylinder. M5-88 TOPIC 2: Volume of Curved Figures
2 Consider the cylinder shown. The radius of the circular base is 5 units and the height of the cylinder is 8 units Suppose there is a circular disc of height 1 unit at the bottom of the cylinder. a. Calculate the area of the top of the circular disc. Recall these formulas for circles. A 5 pr 2 C 5 2pr b. How many congruent circular discs would fill the cylinder? What is the volume of each disc? Explain your reasoning. c. Determine the total volume of the cylinder. Explain your strategy. 4. Write a formula for the volume of a cylinder, where V represents the volume of the cylinder, r represents the radius of the cylinder, and h represents the height of the cylinder. How is this formula like the volume formula for prisms? LESSON 1: Drum Roll, Please! M5-89
3 ACTIVITY 1.2 Volume Problems The director of the marketing department at the Rice Is Nice Company sent a memo to her product development team. She requested that the volume of the new cylinder prototype equal cm Two members of the marketing team claim to have created appropriate prototypes, but they disagree about the dimensions of the cylinder prototype. Cassandra designed the cylinder prototype on the left, and Robert designed the cylinder prototype on the right. Who is correct? What would you say to Cassandra and Robert to settle their disagreement? 4 cm Rice Is Nice 12 cm Rice Is Nice 8 cm 3 cm M5-90 TOPIC 2: Volume of Curved Figures
4 Use what you know about cylinders to solve real-world problems. 2. A circular swimming pool has a diameter of 30 feet and a height of 5 feet. What is the volume of the pool? 3. How many milliliters of liquid are needed to fill a cylindrical can with a radius of 3 centimeters and a height of 4.2 centimeters? One milliliter is equivalent to one cubic centimeter of liquid. 4. Many newspapers are made from 100% wood. The wood used to make this paper can come from pine trees, which are typically about 60 feet tall and have diameters of about 1 foot. However, only about half of the volume of each tree is turned into paper. Suppose it takes about 0.5 cubic inch of wood to make one sheet of paper. About how many sheets can be made from a typical pine tree? Show your work, and explain your reasoning. 5. The volume of each solid is 500 cm 3. Calculate the unknown length in each figure. a. r b. 3 cm 13 cm h 4.5 cm c. d 5 cm LESSON 1: Drum Roll, Please! M5-91
5 ACTIVITY 1.3 Doubling Dimensions Juan and Sandy are discussing the effect that doubling the length of the radius of the base has on the volume of a cylinder. 1. Juan insists that if the length of the radius of a cylinder doubles, the volume will double. Sandy thinks the volume will be more than double. Who is correct? Explain your reasoning. M5-92 TOPIC 2: Volume of Curved Figures
6 2. Sandy and Juan wondered if the results from Question 1 were the same, regardless of the numbers they used. They created a table, hoping to see a pattern. Complete the table, and identify any patterns that you notice. Radius (cm) Area of Base (cm) Height (cm) Volume (cm 3 ) The pattern is easier to recognize if you leave your answers in terms of pi Juan and Sandy are also interested in the effect that doubling the height has on the volume of a cylinder. Juan insists that if the height of a cylinder doubles, the volume will double. Sandy says the volume again will be more than double. Who is correct? Explain your reasoning. 4. Explain why the effect of doubling the length of the radius is different from the effect of doubling the height. LESSON 1: Drum Roll, Please! M5-93
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