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13.3 Volumes of spheres

13.3 Volumes of spheres. By Kylie Bolton and Jessica Nguyen. Objectives. Find volumes of spheres and hemispheres. Solve problems involving volumes of spheres and hemispheres with given diameters, radius’, surface areas, and circumferences. What is a Sphere?.

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13.3 Volumes of spheres

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  1. 13.3 Volumes of spheres By Kylie Bolton and Jessica Nguyen

  2. Objectives • Find volumes of spheres and hemispheres. • Solve problems involving volumes of spheres and hemispheres with given diameters, radius’, surface areas, and circumferences.

  3. What is a Sphere? • A sphere is the set of points in space equidistant from a certain point. • It has no faces, edges, or vertices. • An example is our earth.

  4. How do we get the equation for the volume of a sphere? Finding the volume of a sphere is kind of like finding the volume of a pyramid and the surface area of a sphere. Suppose that the space inside a sphere is divided into an infinite number of near pyramids with all their vertices on the center of the sphere. The height of the pyramid equals the radius of the sphere. The sum of all the pyramids’ bases will equal the surface area of the entire sphere. The sum of all the pyramids’ volume is equal to the volume of the sphere.

  5. Recall, V(pyramid)=1/3Bh, where B is the area of its base and h is its height. Also, the surface area of a sphere, 4∏r², is equal to B1 + B2 + B3 +…+ Bn Sum of the volumes of all the pyramids. Replace h with r. Distributive Property Replace B1+B2+B3+…+Bn With 4∏r² Simplify V=1/3B¹h¹ + 1/3B²h² + 1/3B³h³ +…+ 1/3Bnhn =1/3B1r + 1/3B2r + 1/3B3r +…+ 1/3Bnr =1/3r(B1 +B2 + B3 +…+ Bn) =1/3r(4∏r²) =4/3∏r³

  6. Volume of a Sphere • If a sphere has a volume of a V cubic units and a radius of r units, then V=4/3r³

  7. Find r C=2∏r 24=2∏r R≈3.82m Plug it in! V=4/3∏(3.82)³ V=233.5m³ • Find the Volume of the sphere if C=24m • Round to the nearest 10th

  8. Example 2: • Billy was delivering spherical teapots. He accidently spilt the tea out of one. So he needs to know how much tea to put in the teapots. • Help him out by finding the volume of the spherical teapot if the radius=12in. • Round to the nearest 10th. 12

  9. Continued • Find the volume of the jug. • V=4/312³ • V=7238.2in³ •  • Substitution • Multiply

  10. Volume of a Hemisphere • Hemisphere- half of a sphere. • V=1/2(4/3r³)

  11. Example: Find the volume of this hemisphere with a radius of 2in. 2in

  12. Continued 1. Plug the radius into the formula V=1/2(4/3r³) 2. V=1/2(4/32³) 3. V=1/2(33.5) 4. V=16.8

  13. Example 1: Find the volume of the hemisphere if the surface area=10 Round to the nearest 10th

  14. Continued • T=1/2(4r²)+ r² • 10=1/2(4r²)+ r² • 10=3r² • 3.333=r² • 1.83=r • V=1/2(4/3r³) • V=1/2(4/31.83³) • V=12.8 units² • Surface area of a sphere • Substitute • Divide both sides by 3∏ • Find the sq. root • Volume of a sphere • Substitute • Use a calculator

  15. Assignment Pg. 704 #3-7, 9-20, 23-24, 30-31 By Jessica and Kylie!

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