At 9:00 am, here's what we know about two airplanes: Airplane #1 has an elevation of 51920 ft and is descending at the rate of [tex]$850 ft / min$[/tex]. Airplane #2 has an elevation of 7320 ft and is climbing at the rate of [tex]$150 ft / min$[/tex].
(1) Let [tex]$t$[/tex] represent the time in minutes since 9:00 am, and let [tex]$E$[/tex] represent the elevation in feet. Write an equation for the elevation of each plane in terms of [tex]$t$[/tex].
plane #1: [tex]$E(t)= \square$[/tex]
plane #2: [tex]$E(t)= \square$[/tex]
(2) At what time will the two airplanes have the same elevation?
[tex]$t= \square$[/tex] minutes after 9:00 am
(3) What is the elevation at that time?
[tex]$E= \square$[/tex] feet
Answer (2)
Establish elevation equations for both airplanes: Airplane #1 descends: E 1 ( t ) = 51920 − 850 t , Airplane #2 climbs: E 2 ( t ) = 7320 + 150 t .
Equate the two elevation equations to find the time when they are at the same elevation: 51920 − 850 t = 7320 + 150 t .
Solve for t : t = 44.6 minutes.
Calculate the elevation at t = 44.6 minutes: E = 14010 feet. The final answer is 14010 .
Explanation
Problem Analysis Let's analyze the problem. We have two airplanes with different initial elevations and rates of ascent/descent. We need to find equations for their elevations as functions of time, determine when they have the same elevation, and find that elevation.
Finding Elevation Equations (1) For Airplane #1, the initial elevation is 51920 ft, and it's descending at 850 ft/min. So, the equation for its elevation E 1 ( t ) is: E 1 ( t ) = 51920 − 850 t For Airplane #2, the initial elevation is 7320 ft, and it's climbing at 150 ft/min. So, the equation for its elevation E 2 ( t ) is: E 2 ( t ) = 7320 + 150 t
Setting Elevations Equal (2) To find the time when the two airplanes have the same elevation, we set the two equations equal to each other: 51920 − 850 t = 7320 + 150 t
Solving for Time Now, let's solve for t :
51920 − 7320 = 150 t + 850 t 44600 = 1000 t t = 1000 44600 = 44.6 So, t = 44.6 minutes.
Finding the Elevation (3) To find the elevation at that time, we substitute t = 44.6 into either equation. Let's use the equation for Airplane #2: E 2 ( 44.6 ) = 7320 + 150 ( 44.6 ) E 2 ( 44.6 ) = 7320 + 6690 = 14010 So, the elevation at that time is 14010 feet.
Examples
Understanding how to model the elevation of airplanes can be applied to various real-world scenarios. For example, air traffic controllers use similar models to ensure safe separation between aircraft. Also, weather forecasting models use similar equations to predict the altitude of weather balloons. Moreover, in drone delivery services, understanding the ascent and descent rates is crucial for efficient route planning and package delivery.
The elevation equations for the airplanes are E 1 ( t ) = 51920 − 850 t and E 2 ( t ) = 7320 + 150 t . They will have the same elevation at t = 44.6 minutes after 9:00 am, and the elevation at that time will be 14010 feet. The process involves setting the equations equal and solving for time, then substituting back to find the elevation.
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