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Activity (Hands-On)Grades 10 - 12

Volumes of Complex Solids

Challenged with a hypothetical engineering work situation in which they need to figure out the volume and surface area of a nuclear power plant’s cooling tower (a hyperbolic shape), students learn to calculate the volume of complex solids that can be classified as solids of revolution or solids with known cross sections. These objects of complex shape defy standard procedures to compute volumes. Even calculus techniques depend on the ability to perform multiple measurements of the objects or find functional descriptions of their edges. During both guided and independent practice, students use (free GeoGebra) geometry software, a photograph of the object, a known dimension of it, a spreadsheet application and integral calculus techniques to calculate the volume of complex shape solids within a margin of error of less than 5%—an approach that can be used to compute the volumes of big or small objects. This activity is suitable for the end of the second semester of AP Calculus classes, serving as a major grade for the last six-week period, with students’ project results presentation grades used as the second semester final test.

A photograph shows three wooden finials of varying sizes; they look somewhat like chess pieces. A finial is a distinctive ornament at the apex of a roof, pinnacle, canopy or similar structure in a building or object (such as a lamp).How do you calculate the volume of a finial?

Problem solving within the constraint of available resources is an important engineering skill. In this activity, by using simple resources (digital photography and free geometry software), students create a blueprint (a good side-view photo) of a complex solid of revolution, and on this blueprint—using an appropriate scale factor—estimate its dimensions, which can be used to calculate volume and/or surface area. Knowing the volume of an object and its material’s density, the object’s weight can be determined. Using this approach, engineers solve the problem of determining the weight of an object of circular cross section that cannot be put on a scale (Figure 1) with no more information than its material density and its profile image.

Two photographs: (left) People walk by the bronze Tsar Bell in Moscow, Russia, which is more than 20 feet tall. (right) A large spherical stone finial on the gatepost of a Georgian house in Aberdeenshire, Great Britain.Figure 1. Since using a scale is difficult to impossible, how might you determine the weights of these huge and heavy objects with circular cross sections that can be considered solids of revolution?

After this activity, students should be able to:

  • Use technology to create a blueprint for a complex symmetric solid.
  • Use technology to perform indirect measurements of a complex-shaped object.
  • Use scale factors to transform blueprint measurements into real dimensions of an object.
  • Use integral calculus formulas to find the volume of a solid of complex shape that can be classified as a solid of revolution or a solid with known cross section.
  • Use a spreadsheet application to numerically evaluate integrals.
  • Use a spreadsheet application’s graphing capabilities to visualize the revolving line of a solid of revolution.

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