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Geometry

9. Geometry. Ancient and Modern Mathematics Embrace. Volume and Surface Area. 9.4. Understand the idea of volume of basic three-dimensional objects. Be able to apply the volume and surface area formulas for cylinders.

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Geometry

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  1. 9 Geometry Ancient and Modern Mathematics Embrace

  2. Volume and Surface Area 9.4 • Understand the idea of volume of basic three-dimensional objects. • Be able to apply the volume and surface area formulas for cylinders. • Understand the relationship between the volume and surface area formulas for cylinders, spheres, and cones.

  3. Volume We measure the volumeof a three-dimensional figure using cubic units.

  4. Volume

  5. Volume

  6. Volume • Example: Which block contains the greater volume? (continued on next slide)

  7. Volume • Solution: • Trapezoidal Block • Area of base: • Volume: • Triangular Block • Area of base: • Volume: greater volume

  8. Cylinders

  9. Cylinders • Example: An art supply store sells paint solvent in a cylindrical can that has a diameter of 4 inches and a height of 6 inches. Another can of the same solvent has a diameter twice as large (the height of the can is the same) and costs three times as much as the smaller can. Which can is the better deal? (continued on next slide)

  10. Cylinders • Solution: • Smaller Can • Volume: • Larger Can • Volume: • The large can contains four times as much solvent but costs only three times as much.

  11. Cylinders • Example: An art supply store sells paint solvent in a cylindrical can that has a diameter of 4 inches and a height of 6 inches. Another can of the same solvent has a diameter twice as large (the height of the can is the same) and costs three times as much as the smaller can. Compare the surface areas of the two cans. (continued on next slide)

  12. Cylinders • Solution: • Smaller Can • Surface Area: • Larger Can • Surface Area:

  13. Cylinders • Example: What are the dimensions of a can that will contain 1 cubic foot of liquid and that will have the smallest amount of surface area? • Solution: (continued on next slide)

  14. Cylinders • Solution: • Volume: • Surface Area: (continued on next slide)

  15. Cylinders The ideal radius is between 0.5 and 0.6. Continuing we get .

  16. Cones and Spheres

  17. Cones and Spheres • Example: What is the volume and surface area of a cone with a height of 10 meters and a base radius of 8 meters? • Solution: • Volume: • Surface Area:

  18. Cones and Spheres • Example: A state has circular cone sheds for storing salt. The sheds will have a diameter of 30 feet and a height of 10 feet. Proposed larger sheds will either have a 10-foot-longer diameter and the same height, or the same diameter and a 10-foot greater height. Calculate the volume for each of the proposed sheds. (continued on next slide)

  19. Cones and Spheres • Solution: • Volume(diameter increased): • Volume(height increased):

  20. Cones and Spheres • Example: A state has circular cone sheds for storing salt. The sheds will have a diameter of 30 feet and a height of 10 feet. Proposed larger sheds will either have a 10-foot-longer diameter and the same height, or the same diameter and a 10-foot greater height. Calculate the surface area for each of the proposed sheds. (continued on next slide)

  21. Cones and Spheres • Solution: • Surface Area (diameter increased): • Surface Area (height increased):

  22. Cones and Spheres

  23. Cones and Spheres • Example: A city is replacing existing spherical water tanks with larger spherical water tanks. The new capacity of the tanks should be at least five times the capacity of the old water tanks. If the city purchases tanks that have a radius that is twice the radius of the old tanks, will these • tanks satisfy the desired requirement? (continued on next slide)

  24. Cones and Spheres • Solution: • Old Tanks: • New Tanks: • The new tank’s volume will be 8 times that of the old tanks.

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