To determine how many times the graph of the altitude of the plane will touch the horizontal axis during a roundtrip from Chicago to New York, we can analyze the flight path considering the altitude transitions during takeoff, cruising, landing, and stops.
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Takeoff from Chicago: The plane starts at ground level (altitude = 0) at Chicago and ascends to cruising altitude. This counts as one interaction with the horizontal axis (when the plane takes off).
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Intermediate Stops on the Way to New York:
- The plane has 3 intermediate stops on the way to New York. The sequence of events for each stop is:
- Descent to land (altitude = 0) at each stop.
- Takeoff again to reach cruising altitude.
- Each stop involves touching the horizontal axis twice (once for landing and once for taking off).
- Therefore, for 3 stops, this adds \(3 , \text{stops} \times 2 = 6\) touches to the horizontal axis.
- The plane has 3 intermediate stops on the way to New York. The sequence of events for each stop is:
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Landing in New York: Upon reaching New York, the plane descends and lands, touching the horizontal axis again. This adds one more touch.
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Return Trip from New York to Chicago: The plane will follow a similar pattern returning to Chicago:
- Starts at ground level in New York, thus touching the horizontal axis during takeoff.
- There is 1 intermediate stop on the way back:
- Landing at the stop (touching the horizontal axis once) and taking off again (touching the horizontal axis again). This counts for 2 more touches.
- Finally, the plane lands in Chicago, adding yet another touch.
Now, let's sum all these interactions with the horizontal axis:
- Takeoff from Chicago: 1 touch
- 3 intermediate stops to New York: 6 touches
- Landing in New York: 1 touch
- Takeoff from New York: 1 touch
- 1 intermediate stop on the way back: 2 touches
- Landing in Chicago: 1 touch
Adding these together:
\[ 1 + 6 + 1 + 1 + 2 + 1 = 12 \]
Thus, the graph will touch the horizontal axis a total of 12 times.