Unit 26: Non-Isometric Scaling
Practice Problems
Note: problem difficulty is ranked using a star system.
(*) One-star problems are fundamental to the unit, and can be done relatively quickly. Use these problems to introduce yourself to the material.
(**) Two-star problems are more difficult, and require an understanding of one or two key concepts. Use these problems to test your understanding of the material.
(***) Three-star problems are the most difficult, and require some creative thinking in addition to a deep familiarity with multiple key concepts. Use these problems to challenge yourself; if you can complete one of these, you’re on your way to mastering the material.
**Q26.1) A pizza joint is offering a medium 12-inch pizza for $10 and a large 18-inch pizza for $20. As a student, you’re of course interested in maximizing the amount of pizza you receive per dollar spent. Should you buy one large pizza or two medium pizzas? Use a scaling argument to justify your decision.
**Q26.2) Imagine scaling up a crocodile so that all of its body dimensions (excluding the legs) increase in length by a factor of 4. By what factor must the diameter of its legs grow in order to support its new bodyweight (and experience the same stress as the smaller crocodile’s legs)?
**Q26.3) The California condor is the largest species of bird that is able to sustain flight, with an average adult body mass of 10kg. By comparison, the ruby-throated hummingbird has an average adult mass of 3.1g and a wingspan of only 8.8cm. Using a simple scaling argument, estimate the wingspan of the California condor. How does your prediction compare with the true wingspan? If there is a discrepancy, what might account for it? Hint: birds must be able to produce enough lift to support their body mass; the amount of lift that can be generated is proportional to the surface area of their wings.