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Volume and Displacement

Volume and Displacement. “Satisfaction lies in the effort, not in the attainment. Full effort is full victory.” Mohandas K. Gandhi. Objectives. Discover the formula for the volume of a sphere Apply volume formulas to problems involving spheres or hemispheres.

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Volume and Displacement

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  1. Volume and Displacement “Satisfaction lies in the effort, not in the attainment. Full effort is full victory.” Mohandas K. Gandhi

  2. Objectives • Discover the formula for the volume of a sphere • Apply volume formulas to problems involving spheres or hemispheres. • Find volumes of irregularly shaped solids through displacement.

  3. What do we know about spheres? radius great circle center Vocabulary: Radius, Center, Great Circle

  4. What is volume? • Volume is the measure of the amount of ________ contained in a __________. solid space

  5. Volume of a Sphere Investigation: The Formula for the Volume of a Sphere p542 This investigation demonstrates the relationship between the volume of a hemisphere with radius r and the volume of right cylinder with base radius rand height 2r. r 2r r

  6. Investigation: Steps 1 and 2 After you poured the contents of the hemisphere into the cylinder, what fraction of the cylinder does the hemisphere appear to fill?

  7. Investigation: Step 3 • After you filled the hemisphere the second time and pour the contents into the cylinder, what fraction of the cylinder was filled with water from the two hemispheres (or one ___________)? sphere • The volume of the ________ is the volume of the ___________ . sphere cylinder

  8. Investigation: Step 4 • If the radius of the cylinder is r and its height is 2r, then what is the volume of the cylinder in terms of r? r Vcylinder = Base Area * Height V = (r2)2r V = 2r3 2r

  9. Investigation: Step 5 The volume of a sphere is the fraction of the cylinder’s volume that was filled by two hemispheres (or one __________). What is the formula for the volume of a sphere? sphere Remember when… we found the area of a sector? (r2) (2r3) Vsphere= ()(Vcylinder) = ()(2r3) = ()(r3)

  10. Sphere Volume Conjecture Sphere Volume Conjecture The volume of a sphere with radius r is given by the formula ______. ()(r3)

  11. Hemisphere The volume of a hemisphere with radius r is given by the formula: V = ()(2r3) V = (r3) V = ()()(r3) V = (r3) V =(r3)

  12. Objectives • Discover the formula for the volume of a sphere • Apply volume formulas to problems involving spheres or hemispheres. • Find volumes of irregularly shape solids through displacement.

  13. Example Find the volume of plastic (to the nearest cubic inch) need to make this bowl. The outer-hemisphere diameter is 6.0in and the inner-hemisphere diameter is 5.0in. 6 in 4 in Outer Hemisphere V =(r3) V =((3)3) V =(27) V = 2(9) V =18 Inner Hemisphere V =(r3) V =((2)3) V =(8) V = 5.33 V = 18 - 5.33 = 12.67 in3  40 in3

  14. Example Find the area of a sphere whose radius measures 6cm, to the nearest tenth. V = ()(r3) V = ()((6)3) V = ()(216) V = (4)(72) V = 288 cm3  904.8 cm3 6 cm

  15. Objectives • Discover the formula for the volume of a sphere • Apply volume formulas to problems involving spheres or hemispheres. • Find volumes of irregularly shape solids through displacement.

  16. Displacement For a shape for which we have no formula for calculating volume, it is possible to find the volume by submerging the object in a liquid. Think of when you step into a bathtub that is filled to the brim. The volume of the liquid that overflows in each case equals the volume of the solid below the liquid level. This volume is called an object’s ______________. displacement

  17. Example 2 cm 15 cm 15 cm 10 cm 10 cm What is the volume of the rock? V = 10 * 15 * 2 = 300 cm3

  18. Density An important property of a material is its density. Density is the _______ of matter in a given __________. Formula: mass volume

  19. Example A square-prism container with a base 5cm by 5cm is partially filled with water. You drop a clump of metal that weighs 525 g into the container, and the water level rises 2cm. What is the density of the metal? Assuming the metal is pure, what is the metal? Density = Density = = = 10.5 g/cm3 Look at the chart on p535 for the metal name. 2 cm 5 cm 5 cm 5 cm 5 cm Silver

  20. Example A clump of metal weighs 351.4 grams. If the metal is pure silver, which has a density of 10.50 g/cm3, what is the volume of the clump of silver? Round to the nearest tenth. Density = • 10.50 =  Plug in what we know • = Cross Multiply • 10.50(Volume) = 351.4  Divide by 10.50 • Volume = 33.5 cm3

  21. Objectives • Discover the formula for the volume of a sphere • Apply volume formulas to problems involving spheres or hemispheres. • Find volumes of irregularly shape solids through displacement.

  22. Closing The formula for the volume of a sphere is ___________. You can find the __________ of an ___________ shaped object by measuring the amount of liquid it __________. Volumes can be used to find the ___________ of objects and thus identify the materials of which they are made. The formula to find density is _____________. V = r3 volume irregularly displaces density D =

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