To find the rate of change of the altitude of the airplane with respect to time, we first need to establish the linear relationship between altitude (y) and time (x) using the given points.
We have the following data points:
- (1.5, 28,000)
- (3.25, 24,500)
- (6, 19,000)
To find the rate of change, we can calculate the difference in altitude divided by the difference in time between two consecutive points. Let's calculate the rate between the first two points:
From (1.5, 28,000) to (3.25, 24,500):
- Change in altitude = 24,500 - 28,000 = -3,500 feet
- Change in time = 3.25 - 1.5 = 1.75 minutes
Rate of change = Change in altitude / Change in time = -3,500 / 1.75 = -2,000 feet per minute
Now, we can verify that this rate remains constant with the next set of points. Let's calculate the change from (3.25, 24,500) to (6, 19,000):
From (3.25, 24,500) to (6, 19,000):
- Change in altitude = 19,000 - 24,500 = -5,500 feet
- Change in time = 6 - 3.25 = 2.75 minutes
Rate of change = -5,500 / 2.75 = -2,000 feet per minute
Both calculations give us a consistent rate of change of -2,000 feet per minute.
Statement:
The altitude of the airplane is changing at a constant rate of -2,000 feet per minute with respect to time. This indicates that the airplane is descending at a steady rate.
Now, let’s calculate the missing values for altitude corresponding to the time to reach an altitude of 2,000 feet.
Starting from the last known altitude (19,000 feet at 6 minutes) and using the rate of -2,000 feet per minute:
To find the time when altitude reaches 2,000 feet:
- From 19,000 feet to 2,000 feet, the altitude decreases by 17,000 feet.
- Time taken = Change in altitude / Rate of change = 17,000 / 2,000 = 8.5 minutes.
Adding this time to the previous time (6 minutes):
- \(6 + 8.5 = 14.5\).
Completion:
The missing time value when the altitude is 2,000 feet is 14.5 minutes.
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