Student Page for Using Diameter, Volume and Surface Area to Determine the Dimensions of PSA's Computer

Introductory video

The Main Problem:
NASA needs you to design a computer for the Personal Satellite Assistant, a robot helper that will assist the astronauts in space. This robot is shaped like a sphere and has a diameter of 8 inches. The computer must meet the following criteria:

1. a rectangular prism with a volume of 24 cubic inches
2. must fit inside a sphere with a diameter of 8 inches
3. must have the largest possible surface area to allow the computer to easily release heat so that it doesn't overheat

Main Problem video

Prerequisites:

• What is diameter?
• What is diameter of a sphere?
• How do you calculate volume?
• What is surface area?
• How do you calculate surface area?

Prerequisites video

• Multiplying and dividing decimals.
• A rectangular prism is a three dimensional solid with six rectangular faces.
• A sphere is a round three-dimensional solid with all points of the surface at the same distance from the center.

Materials:
For each group:

• 24 one-inch cubes
• 1 paper circle with an 8-inch diameter
• graph paper
• 1 calculator per student

Procedure

1. Use your materials to find solutions to your problem.
2. Check that each solution meets the criteria in the problem.
3. Questions to consider as you look for solutions:
   • How many possible solutions do you think there are?
   • Does a solution have to have all whole numbers for the volume dimensions?
   • How small does the width of the computer need to be to fit within the curvature of the sphere so that no corners stick out?
   • What causes surface area to increase?
   • What shapes have the greatest surface area?
   • As the height of the computer decreases, what happens to the surface area?
   • How do you know that your answer is the best answer?
4. Write a letter to NASA with your recommendation of the dimensions the computer should be, how you figured it out and why you think it's the best answer.

Extension Problem
NASA’s robot also uses fans to move around in microgravity. These fans are shaped like cylinders. The cylinders also need to have the maximum surface area in order to release heat. Should NASA use short and fat cylinder fans or long and skinny cylinder fans? Explain your answer.

Extension Problem video

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