According to the Travel Almanac, the world’s fastest land animal is the cheetah. It can travel at up to 70 mph. Think of this scenario: A cheetah sitting under a tree sprints toward its prey at 70 mph. It runs back to its initial spot by the tree at a modest 40 mph. The cheetah has embarked on a round-trip. 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?
That’s one of the things you’ll determine as you work to complete this task. Make a conjecture. What do you think the answer will be?

Suppose the cheetah sprinted at maximum speed for 8 minutes and then slowed to 40 mph for the next 8 minutes.
How far would the cheetah have traveled in the first 8 minutes? Show how you arrived at your answer.

-> 9.3336 miles

How far would the cheetah have traveled in the next 8 minutes? Show how you arrived at your answer.
-> 5.3 miles

How much farther did the cheetah travel in the first 8 minutes than in the second 8 minutes?
-> 4.0336 miles

The cheetah traveled 1.75 times faster for the first 8 minutes than it did for the second 8 minutes. Was the distance traveled during the first 8 minutes 1.75 times greater than the distance traveled during the second 8 minutes? Show the calculation to justify your answer.

If the cheetah made a round-trip and took half the amount of time on the return trip as on the front end of the trip, what would be the relationship between the average rates on each leg of the trip? Using a complete sentence, explain how you arrived at this conclusion.

Step 3: Calculating time
Assume that the cheetah travels an average of 40 mph to go from its resting place to a rock near a river. On the return trip to its resting place, the cheetah travels an average of 70 mph. If the cheetah traveled for 15 minutes, how many minutes did the return trip take to the nearest minute and second?
Set up the table as follows. Label the rows “To the River” and “From the River.” Label the columns “Distance,” “Rate,” and “Time (in Hours).” Let t represent the unknown quantity in the problem. Fill in the table.

From the table, set up an equation relating the distances.

Solve the problem. Write the answer in a complete sentence, stating it in terms of minutes and seconds.

Now it’s time to tackle the original question.
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?
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: d=r_1 t_1
Second leg of trip: d=r_2 t_2
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.

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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.

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.

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