16.1 Volume of Prisms and Cylinders

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1 Name Class Date 16.1 Volume of Prisms and Cylinders Essential Question: How do te formulas for te volume of a prism and cylinder relate to area formulas tat you already know? Explore G.11.D Apply te formulas for te volume of tree-dimensional figures, including prisms, cylinders, to solve problems using appropriate units of measure. Developing a Basic Volume Formula Resource Locker Te volume of a tree-dimensional figure is te number of nonoverlapping cubic units contained in te interior of te figure. Tis prism is made up of 8 cubes, eac wit a volume of 1 cubic centimeter, so it as a volume of 8 cubic centimeters. You can use tis idea to develop volume formulas. Volume = 1 cubic unit In tis activity you ll explore ow to develop a volume formula for a rigt prism and a rigt cylinder. A rigt prism as lateral edges tat are perpendicular to te bases, wit faces tat are all rectangles. A rigt cylinder as bases tat are perpendicular to its center axis. axis rigt prism rigt cylinder B On a seet of paper draw a quadrilateral sape. Make sure te sides aren t parallel. Assume te figure as an area of B square units. Use it as te base for a prism. Take a block of Styrofoam and cut to te sape of te base. Assume te prism as a eigt of 1 unit. How would canging te area of te base cange te volume of te prism? area is B square units eigt is 1 unit Module Lesson 1

2 If te base as an area of B square units, ow many cubic units does te prism contain? Now use te base to build a prism wit a eigt of units. eigt is units How muc greater is te volume of tis prism compared to te one wit a eigt of 1? Reflect 1. Suppose te base of te prism was a rectangle of sides l and w. Write a formula for te volume of te prism using l, w, and. 2. A cylinder as a circular base. Use te results of te Explore to write a formula for te volume of a cylinder. Explain wat you did. Explain 1 Finding te Volume of a Prism Te general formula for te volume of a prism is V = B. Wit certain prisms te volume formula can include te formula for te area of te base. Volume of a Prism Te formula for te volume of a rigt rectangular prism wit lengt l, widt w, and eigt is V = lw. Te formula for te volume of a cube wit edge lengt s is V = s. S l S W S Module Lesson 1

3 Example 1 Use volume formulas to solve real world problems. A sark and ray tank at te aquarium as te dimensions sown. Estimate te volume of water in gallons. Use te conversion 1 gallon = 0.14 f t. 120 ft Step 1 Find te volume of te aquarium in cubic feet. V = lw = (120) (60) (8) = 57,600 f t 8 ft Step 2 Use te conversion factor _ 1 gallon to estimate 0.14 ft te volume of te aquarium in gallons. 60 ft 1 gallon 57,600 ft _ 0.14 ft 429,851 gallons _ 1 gallon 0.14 ft = 1 Step Use te conversion factor 429,851 gallons 1 gallon to estimate te weigt of te water. 8. pounds 8. pounds,580,659 pounds 1 gallon 8. pounds = 1 1 gallon Te aquarium olds about 429,851 gallons. Te water in te aquarium weigs about,580,659 pounds. B Cemistry Ice takes up more volume tan water. Tis cubic container is filled to te brim wit ice. Estimate te volume of water once te ice melts. Density of ice: g/c m Density of water: 1 g/ cm Step 1 Find te volume of te cube of ice. V = s = = c m Step 2 Convert te volume to mass using te conversion factor g_ cm. c m g_ c m g cm Step Use te mass of ice to find te volume of water. Use te conversion factor. Reflect 24.8 g c m. Te general formula for te volume of a prism is V = B. Suppose te base of a prism is a parallelogram of lengt l and altitude. Use H as te variable to represent te eigt of te prism. Write a volume formula for tis prism. Module Lesson 1

4 Your Turn 4. Find te volume of te figure. 5. Find te volume of te figure. Volume = 8 cubic units Eac cube as a side of 2k. 6. Wat is te area of te base of te pool? Recall tat 0.14 f t _ 1 gallon. 10 ft V = 50,000 gallons Explain 2 Finding te Volume of a Cylinder You can also find te volume of prisms and cylinders wose edges are not perpendicular to te base. Oblique Prism An oblique prism is a prism tat as at least one non-rectangular lateral face. Oblique Cylinder An oblique cylinder is a cylinder wose axis is not perpendicular to te bases. Module Lesson 1

5 Cavalieri s Principle If two solids ave te same eigt and te same cross-sectional area at every level, ten te two solids ave te same volume. Example 2 To find te volume of an oblique cylinder or oblique prism, use Cavalieri s Principle to find te volume of a comparable rigt cylinder or prism. Te eigt of tis oblique cylinder is tree times tat of its radius. Wat is te volume of tis cylinder? Round to te nearest tent. Use Cavalieri s Principle to find te volume of a comparable rigt cylinder. Represent te eigt of te oblique cylinder: = r Use te area of te base to find r. π r 2 = 81π cm 2, so r = 9 cm. Calculate te eigt: = r = 27 cm Calculate te volume. V = B = (81π) Te volume is about cubic centimeters. B = 81π cm 2 Module Lesson 1

6 B Te eigt of tis oblique square-based prism is four times tat of side lengt of te base. Wat is te volume of tis prism? Round to te nearest tent. Calculate te eigt of te oblique prism: = s, were s is te lengt of te square base. Use te area of te base to find s. s 2 = c m 2 s = _ cm Calculate te eigt. = 4s = 4 cm Calculate te volume. V = B = (75 c m 2 ) ( = c m cm) B = 75 cm 2 Your Turn Find te volume r = 12 in. = 45 in. = (x + 2) cm 4x cm 5x cm Explain Finding te Volume of a Composite Figure Recall tat a composite figure is made up of simple sapes tat combine to create a more complex sape. A composite tree-dimensional figure is formed from prisms and cylinders. You can find te volume of eac separate figure and ten add te volumes togeter to find te volume of te composite figure. Module Lesson 1

7 Example Find te volume of eac composite figure. 5 ft Find te volume of te composite figure, wic is an oblique cylinder on a cubic base. Round to te nearest tent. Te base area of te cylinder is B = π r = π (5) = 25π ft. Te cube as side lengts equal to te diameter of te cylinder s circular base: s = 10. Te eigt of te cylinder can now be calculated: = = 12 ft. Te volume of te cube is V = s = 10 = 1000 f t. Te volume of te cylinder is V = B = (25π ft 2 ) (12 ft) f t. Te total volume of te composite figure is te sum of te individual volumes. V = 1000 f t f t = f t = 22 ft B Tis periscope is made up of two congruent cylinders and two congruent triangular prisms, eac of wic is a cube cut in alf along one of its diagonals. Te eigt of eac cylinder is 6 times te lengt of te radius. Use te measurements provided to estimate te volume of tis composite figure. Round to te nearest tent. Use te area of te base to find te radius. B = π r 2 π r 2 = π B = 6π in 2 r = in. Calculate te eigt of eac cylinder: = 6r = 6 = in. Te faces of te triangular prism tat intersect te cylinders are congruent squares. Te side lengt s of eac square is te same as te diameter of te circle. s = d = 2 = in. Te two triangular prisms form a cube. Wat is te volume of tis cube? V = s = = in Find te volume of te two cylinders. V = 2 6π = in Te total volume of te composite figure is te sum of te individual volumes. V = in + in in Module Lesson 1

8 Reflect 9. A pipe consists of two concentric cylinders, wit te inner cylinder ollowed out. Describe ow you could calculate te volume of te solid pipe. Write a formula for te volume. r 1 r2 \ Your Turn 10. Tis robotic arm is made up of two cylinders wit equal volume and two triangular prisms for a and. Te volume of eac prism is 1 2 r 1 r 2r, were r is te radius of te cylinder s base. Wat fraction of te total volume does te and take up? r Elaborate 11. If an oblique cylinder and a rigt cylinder ave te same eigt but not te same volume, wat can you conclude about te cylinders? 12. A rigt square prism and a rigt cylinder ave te same eigt and volume. Wat can you conclude about te radius of te cylinder and side lengts of te square base? 1. Essential Question Ceck-In How does te formula for te area of a circle relate to te formula for te volume of a cylinder? Module Lesson 1

9 Evaluate: Homework and Practice 1. Te volume of prisms and cylinders can be represented wit B, were B represents te area of te base. Identify te type of figure sown and matc te prism or cylinder wit te appropriate volume formula. Online Homework Hints and Help Extra Practice A. V = (π r 2 ) B. V = ( 1_ 2 b ) C. V = lw Find te volume of eac prism or cylinder. Round to te nearest undredt mm 8.4 mm.5 mm 15 yd 9 yd 12 yd 4. Te area of te exagonal base is ( 54 tan 0 ) m Te area of te pentagonal base is ( 125 tan 6 ) m 2. Its eigt is 8 m. Its eigt is 15 m cm 10 ft 4 cm 9 cm 12 ft Module Lesson 1

10 8. Multi-Step A vase in te sape of an oblique cylinder as te dimensions sown. Wat is te volume of te vase in liters? Round to te nearest tundredt. (Hint: Use te rigt triangle in te cylinder to find its eigt.) 17 cm 14 cm Find te volume of eac composite figure. Round to te nearest tent ft in. 5 in. 4 ft 4 ft 14 ft 15 in. 12 ft cm 12. Te two figures on eac end 4 cm 4 cm combine to 6 cm form a rigt cylinder. 6 cm 6 cm 8 cm 8 cm 8 cm 4 ft 2 ft 12 ft 4 ft 2 ft Module Lesson 1

11 1. Colin is buying dirt to fill a garden bed tat is a 9 ft by 16 ft rectangle. If e wants to fill it to a dept of 4 in., ow many cubic yards of dirt does e need? Round to te nearest cubic yard. If dirt costs $25 per y d, ow muc will te project cost? 14. Persevere in Problem Solving A cylindrical juice container wit a in. diameter as a ole for a straw tat is 1 in. from te side. Up to 5 in. of a straw can be inserted. a. Find te eigt of te container to te nearest tent. 5 in. 1 in. in. b. Find te volume of te container to te nearest tent. c. How many ounces of juice does te container old? (Hint: 1 i n 0.55 oz) Module Lesson 1

12 15. Abigail as a cylindrical candle mold wit te dimensions sown. If Abigail as a rectangular block of wax measuring 15 cm by 12 cm by 18 cm, about ow many candles can se make after melting te block of wax? Round to te nearest tent. 6.0 cm.4 cm 16. A 96-inc piece of wire was cut into equal segments tat were ten connected to form te edges of a cube. Wat is te volume of te cube? 17. Algebra Find te volume of te tree-dimensional figure in terms of x. x + 1 x 18. One cup is equal to i n. If a 1-cup measuring cylinder as a radius of 2 in., wat is its eigt? If te radius is 1.5 in., wat is its eigt? Round to te nearest tent. Module Lesson 1

13 19. Make a Prediction A cake is a cylinder wit a diameter of 10 in. and a eigt of in. For a party, a coin as been mixed into te batter and baked inside te cake. Te person wo gets te piece wit te coin wins a prize. a. Find te volume of te cake. Round to te nearest tent. b. Keka gets a piece of cake tat is a rigt rectangular prism wit a in. by 1 in. base. Wat is te probability tat te coin is in er piece? Round to te nearest undredt. H.O.T. Focus on Higer Order Tinking 20. Multi-Step Wat is te volume of te tree-dimensional object wit te dimensions sown in te tree views? 4 cm 4 cm 10 cm 10 cm 10 cm 10 cm Front Top Side 21. Draw Conclusions You can use displacement to find te volume of an irregular object, suc as a stone. Suppose a 2 foot by 1 foot tank is filled wit water to a dept of 8 in. A stone is placed in te tank so tat it is completely covered, causing te water level to rise by 2 in. Find te volume of te stone. Module Lesson 1

14 22. Analyze Relationsips One juice container is a rectangular prism wit a eigt of 9 in. and a in. by in. square base. Anoter juice container is a cylinder wit a radius of 1.75 in. and a eigt of 9 in. Describe te relationsip between te two containers. Lesson Performance Task A full roll of paper towels is a cylinder wit a diameter of 6 inces and a ollow inner cylinder wit a diameter of 2 inces. 2 in. 2 in. 2 in. 1. Find te volume of te paper on te roll. Explain your metod. 2. Eac seet of paper on te roll measures 11 inces by 11 inces by 2 1 inc. Find te volume of one seet. Explain ow you found te volume. 11 in.. How many seets of paper are on te roll? Explain. Module Lesson 1