Question 19.46

16. You may remember having to work problems like, “If Joe can dig a ditch in 3 days, and Sam can dig it in 4, how long will it take the two of them working together?” The answer is related to the harmonic mean of 3 and 4. The Environmental Protection Agency uses the harmonic mean to calculate the “average” fuel economy of the fleet of cars from a manufacturer. The formula for the harmonic mean of two numbers and is

  1. Calculate the answer for Joe and Sam, which is one-half of the harmonic mean of 3 and 4. Explain why this is the correct answer. (In terms of fuel economy, using the harmonic mean assumes that each car is driven the same number of miles; using the arithmetic mean would assume that each car uses the same amount of fuel.)
  2. image Show that the harmonic mean of two positive numbers is always less than or equal to the geometric mean. (Thus, in light of Exercise 15, we have the general conclusion that , where stands for the harmonic mean, for the geometric mean, and for the arithmetic mean.) (Hint: Suppose that the claim is false. Simplify the fraction that is the harmonic mean, square both sides of the inequality, and proceed as in Exercise 15.)
  3. image Show once more that the harmonic mean of two positive numbers is always less than the geometric mean, but this time do it with less work: Let and , and discover one connection (equation) between the harmonic mean of and and the arithmetic mean of and , and a second connection between the geometric mean of and and the geometric mean of and . Then use Exercise 15 on and .
  4. image What should be the formula for the harmonic mean of three numbers? Of numbers?

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