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Math 10C Module 1 Lesson 6

This photograph shows a cylindrical propane storage tank with hemispherical ends.

© Jim Parkin/shutterstock

As you look around your surroundings, you may find objects that resemble the 3-D objects studied in the previous two lessons. A soup can, a box, and a ball are examples of cylinders, prisms, and spheres, respectively. A pylon used to alert motorists of traffic obstructions resembles a cone, and some games use pyramid-shaped dice.

While there are many examples of prisms, pyramids, cylinders, spheres, and cones, you may notice that many other objects are actually composites of two or more of these basic 3-D objects.

The image of the industrial propane tank is an example of a composite figure. Can you tell what 3-D objects are used in the tank’s design?

Glossary Terms

Throughout Module 1, you have been adding and saving math terms to “Glossary Terms” in your binder.

In this lesson the suggested terms for your glossary are

  • composite figure
  • hemispherical
  • surface area to volume ratio

Return your updated “Glossary Terms” to your course folder (binder).

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Tip

In complex problems that have more than one three-dimensional shape, it is a good strategy to break the problem into parts so you are dealing with only one shape at a time.

In the case of the propane storage tank shown in the photo, you may want to determine its volume by first determining the volume of the cylinder and then determining the volume of the hemispheres (or half-spheres) on each end.

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The steps below will help you to solve a surface area or volume problem:

Step 1: Decide which 3-D object or objects can be used to model the problem.

Step 2: Draw a rough sketch of this 3-D object, and label its dimensions.

Step 3: Decide which formulas you will use. You may need to select more than one formula in the case of composite figures. You might also find that you will need to modify the formula to fit the problem.

Step 4: Substitute given values into your formulas to solve the problem.

Read

Work through the following textbook examples that show how problems involving composite figures are solved. In the solutions, pay attention to

  • the importance of an accurate drawing
  • any preliminary calculations that need to be made
  • the sequence of steps

Read “Example 1: Determining the Volume of a Composite Object” on pages 56 and 57 in your textbook. Then read “Example 2: Determining the Surface Area of a Composite Object” on page 57. Finally, read “Example 3: Solving a Problem Related to a Composite Object” on page 58.

Self-Check

The photograph shows three round grain storage bins.

© Sue Smith/shutterstock

SC 1. A grain storage bin has a diameter of 4.8 m. The height of the straight side wall is 10 m. The cone top has an additional height of 1.5 m.

  1. What is the total volume of this bin?
  1. Suppose you want to paint the outside of one of the grain storage bins. If the slant height of the conical roof is 2.83 m, then what total surface area needs to be painted?
This graphic shows an illustration of a sphere with a radius of 3.5 cm.

SC 2. Mr. Vanilla charges 0.5 cents/cm3 for his ice cream. How much would you pay for one spherical-shaped scoop of ice cream if a scoop of ice cream has a radius of 3.5 cm?

  1. $8.99
  2. $1.50
  3. 26 cents
  4. 90 cents

SC 3. The right cylinder and right cone shown have the same radius and volume. The cylinder has a height of 12 in. What is h, the height of the cone?

This graphic shows an illustration of a cylinder with radius, r, a height of 12 in, and a cone with radius, r, and height, h.

  1. 18 in
  2. 24 in
  3. 36 in
  4. 42 in

SC 4. A glass containing water is in the shape of a right circular cylinder with a radius of 3 cm. The height of the water in the glass is 10 cm.

  1. What is the volume of the water in the glass? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
  2. Five spherical marbles of equal size are dropped into the glass. The water in the glass rises to a height of 11 cm. What is the increase in the volume of the glass contents? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
  3. What is the volume of one marble? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.
  4. What is the radius of one marble? Be sure to include units of measure in your answer. Show or explain how you obtained your answer.

Compare your answers.

Try This

How did you do on the Self-Check questions?Complete TT1 for more practice:

Foundations and Pre-calculus Mathematics 10 (Pearson)

TT 1. Complete “Exercises” questions 3,  and 9 on pages 59 and 60.

Use the link below to check your answers to Try This 1.

Possible TT1 Solutions

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Caution

There are a few myths about surface area and volume you may hear. These are all INCORRECT.

  • Surface area can be found only for two-dimensional objects.
  • Surface area is a concept used only in mathematics classes.
  • You cannot measure the volume of some objects because they do not have “regular” lengths, widths, or heights.
  • An object’s volume is greater in water than in air.
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