11.

Question

A plane that carries mail makes a round trip each day from Chicago to New York. It makes 3 intermediate stops on the way to New York and 1 intermediate stop on the way back to Chicago. Suppose you make a graph of the altitude of the plane for one day, with time on the horizontal axis and altitude on the vertical axis. How many times will the graph touch the horizontal axis?

Answer 1

To determine how many times the altitude graph of the plane will touch the horizontal axis, we need to consider how the plane's altitude changes during its round trip from Chicago to New York and back.

1. Flight from Chicago to New York:
   • The plane starts at an altitude of 0 (takeoff from Chicago).
   • It ascends to cruising altitude (let's say it reaches a peak altitude somewhere along the journey).
   • It then makes 3 intermediate stops. At each stop, it descends to an altitude of 0 (landing).
   • After each stop, the plane ascends again to reach cruising altitude before continuing to New York.
   • After reaching New York, the plane comes down to an altitude of 0 upon landing.

So for the journey from Chicago to New York:

• Takeoff from 0 to cruising altitude → does not touch the horizontal axis.
• Descends to altitude 0 (touches the horizontal axis once) at the first intermediate stop.
• Ascends back up (does not touch).
• Descends to altitude 0 at the second intermediate stop (touches again).
• Ascends back up (does not touch).
• Descends to altitude 0 at the third intermediate stop (touches again).
• Ascends back up (does not touch).
• Finally descends back down to 0 in New York (touches again).

This gives us a total of 4 touches of the horizontal axis during the trip from Chicago to New York.

2. Flight from New York to Chicago:
   • The plane takes off from New York (altitude 0) and climbs to cruising altitude.
   • Then, it makes 1 intermediate stop on the way back.
   • After making the stop, it descends to 0 (touching the horizontal axis once) upon landing.

So for the journey from New York back to Chicago:

• Takeoff from 0 to cruising altitude → does not touch.
• Descends to altitude 0 at the intermediate stop (touches).

This gives us a total of 2 more touches of the horizontal axis during the trip from New York back to Chicago.

Now, adding both parts together:

• Chicago to New York: 4 touches
• New York to Chicago: 2 touches

In total, the graph will touch the horizontal axis 4 + 2 = 6 times.

Therefore, the answer is 6.