﻿﻿An airplane takes off and lands at another airport an hour later, 400 miles away.
If t represents the time in minutes since the plane has left the airport, let xt) be the horizontal distance traveled and (t) be the altitude of the plane.

﻿﻿Describe xt).

- Describe y (t).

Let's describe the functions \( x(t) \) and \( y(t) \) based on the information about the airplane's flight.

### \( x(t) \) — Horizontal Distance Traveled
- \( x(t) \) represents the horizontal distance the airplane has traveled from its departure point as a function of time \( t \), where \( t \) is in minutes.
- Since the plane travels 400 miles in one hour (60 minutes), \( x(t) \) will start at 0 when \( t = 0 \) (at takeoff) and increase steadily until it reaches 400 miles when \( t = 60 \).
- Assuming constant speed, \( x(t) \) would be a linear function. The speed of the airplane in horizontal distance is:
  \[
  \text{speed} = \frac{400 \, \text{miles}}{60 \, \text{minutes}} = \frac{20}{3} \, \text{miles per minute}
  \]
  Thus, \( x(t) \) can be expressed as:
  \[
  x(t) = \frac{20}{3} t
  \]
  where \( t \) is in minutes and \( x(t) \) is in miles.

### \( y(t) \) — Altitude of the Plane
- \( y(t) \) represents the altitude of the airplane as a function of time \( t \) in minutes.
- The altitude starts at 0 when \( t = 0 \) (the airplane is on the ground at takeoff) and also ends at 0 when \( t = 60 \) (when the airplane lands at the destination).
- The plane ascends during the initial part of the flight, reaching a maximum cruising altitude at some point in the middle of the flight, and then descends back to the ground as it prepares to land.
- Therefore, \( y(t) \) is likely a smooth, bell-shaped curve, possibly modeled by a quadratic or sinusoidal function that reaches a peak altitude at some time between takeoff and landing, say around \( t = 30 \) minutes.

The general shape of \( y(t) \) would be:
- \( y(0) = 0 \) (altitude at takeoff),
- \( y(60) = 0 \) (altitude at landing),
- \( y(t) \) reaches its maximum at some point between \( t = 0 \) and \( t = 60 \), likely near \( t = 30 \).

### Summary:
- \( x(t) \) describes the linear horizontal distance, increasing steadily as \( t \) increases, starting at 0 miles and reaching 400 miles after 60 minutes.
- \( y(t) \) describes the altitude, which follows a smooth curve starting and ending at 0, with a peak altitude somewhere near the middle of the flight.

Do you want more details or have any questions about this?

### 5 Related Questions:
1. How would the functions change if the plane did not travel at a constant horizontal speed?
2. Can you derive a specific function for \( y(t) \), assuming the maximum altitude and time to reach it?
3. How would the function \( x(t) \) change if the distance between airports was longer or shorter?
4. What would be the effect of wind on the horizontal distance function \( x(t) \)?
5. How can we model a realistic acceleration and deceleration for the plane during takeoff and landing?

### Tip:
In real-world applications, airplane velocity is not constant—planes accelerate during takeoff and decelerate during landing. A more realistic model would use piecewise or non-linear functions for both \( x(t) \) and \( y(t) \).

An airplane takes off and lands at another airport an hour later, 400 miles away. If t represents the time in minutes since the plane has left the airport, let x(t) be the horizontal distance traveled and y(t) be the altitude of the plane. Describe x(t) and y(t).

Explore how to model an airplane's horizontal distance traveled and altitude over time. Learn about linear functions for constant velocity and quadratic or sinusoidal functions to represent altitude changes. Suitable for students in Grades 9-12.

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