Step 3a: Calculating the average rate

Going from point A to point B, the cheetah traveled at an average rate of 70 mph. Returning to point A, the cheetah traveled at an average rate of 40 mph.

Can we say that this cheetah's average rate was 55 mph?
a. The average rate for the trip is equal to the total distance traveled divided by the total time traveled. The following equations represent the distance traveled on each leg of the trip.

First leg of trip:
Second leg of trip:

Write an equation for the average rate for the trip. Remember, the cheetah runs from point A to point B and back to point A.

The following equation represents the average rate for the trip. Is this equation equivalent to the one you wrote above? Explain why or why not.

b. The following steps show how the equation for the average rate can be transformed so that it is written in terms of only the rates for each leg of the trip. Write an algebraic justification for each step. Think about the number operations and properties that you know.

c. Using the equation for average rate above, determine the cheetah's average rate for the entire trip.

a) Let AB represent the distance from A to B. Let t1 and t2 represent the travel times (in hours) on leg1 and leg2 of the trip, respectively. Then the distances traveled are...

First leg distance: AB = 70·t1 ⇒ t1 = AB/70

Second leg distance: AB = 40·t2 ⇒ t2 = AB/40

The average rate is the ratio of total distance to total time:

average rate = (AB +AB)/(t1 +t2)

average rate = 2AB/(AB/70 +AB/40) = 2/(1/70 +1/40) = 2(40)(70)/(70+40)

average rate = 560/11 = 50 10/11 . . . mph

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No equations are given, so we cannot compare what we wrote with the given equations. In each step of the solution, we have used the rules of algebra and equality.

b) For two rates over the same distance (as above), the average is their harmonic mean:

average rate = 2r1·r2/(r1+r2)

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c) The cheetah's average rate was 50 10/11 mph ≈ 50.91 mph.