Answer:
Time: 8 minutes
Altitude: 20000ft
Method 1 is easiest
Method 3 is easiest
Step-by-step explanation:
Given
Airplane 1:
![Height = 44800 ft](https://tex.z-dn.net/?f=Height%20%3D%2044800%20ft)
![Descending\ Rate = 3100ft/min](https://tex.z-dn.net/?f=Descending%5C%20Rate%20%3D%203100ft%2Fmin)
Airplane 2:
![Ascending\ Rate = 2500ft/min](https://tex.z-dn.net/?f=Ascending%5C%20Rate%20%3D%202500ft%2Fmin)
Required
Determine when both planes would be at the same altitude?
Let the minute be represented by m
For Airplane 1, Its altitude at any height h is:
![Airplane\ 1 = Height - Descending\ Rate * m](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%20Height%20-%20Descending%5C%20Rate%20%2A%20m)
<em>It is minus (-) because the airplane is descending</em>
![Airplane\ 1 = 44800 - 3100 * m](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%2044800%20-%203100%20%2A%20m)
![Airplane\ 1 = 44800 - 3100m](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%2044800%20-%203100m)
For Airplane 2, Its altitude at any height h is:
![Airplane\ 2 = Ascending\ Rate * m](https://tex.z-dn.net/?f=Airplane%5C%202%20%3D%20Ascending%5C%20Rate%20%2A%20m)
![Airplane\ 2 = 2500 * m](https://tex.z-dn.net/?f=Airplane%5C%202%20%3D%202500%20%2A%20m)
![Airplane\ 2 = 2500m](https://tex.z-dn.net/?f=Airplane%5C%202%20%3D%202500m)
Method 1:
For both heights to be equal, we have that:
![Airplane\ 1 = Airplane\ 2](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%20Airplane%5C%202)
This gives:
![44800 - 3100m = 2500m](https://tex.z-dn.net/?f=44800%20-%203100m%20%3D%202500m)
Collect Like Terms
![44800 = 2500m + 3100m](https://tex.z-dn.net/?f=44800%20%3D%202500m%20%2B%203100m)
![44800 = 5600m](https://tex.z-dn.net/?f=44800%20%3D%205600m)
![5600m = 44800](https://tex.z-dn.net/?f=5600m%20%3D%2044800)
![m = 44800/5600](https://tex.z-dn.net/?f=m%20%3D%2044800%2F5600)
![m = 8\ min](https://tex.z-dn.net/?f=m%20%3D%208%5C%20min)
<em>Hence, the time they will be at the same altitude is 8 minutes</em>
Substitute 8 for m in
![Airplane\ 2 = 2500m](https://tex.z-dn.net/?f=Airplane%5C%202%20%3D%202500m)
![Airplane\ 2 = 2500 * 8](https://tex.z-dn.net/?f=Airplane%5C%202%20%3D%202500%20%2A%208)
![Airplane\ 2 = 20000\ ft](https://tex.z-dn.net/?f=Airplane%5C%202%20%3D%2020000%5C%20ft)
<em>Hence, they will be at the same altitude at 20000ft</em>
Method 2:
We have that:
![Airplane\ 1 = 44800 - 3100m](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%2044800%20-%203100m)
![Airplane\ 2 = 2500m](https://tex.z-dn.net/?f=Airplane%5C%202%20%3D%202500m)
Since they are to be at the same altitude, then
The difference in their altitude must be 0
i.e.
![Airplane\ 1 - Airplane\ 2 = 0](https://tex.z-dn.net/?f=Airplane%5C%201%20-%20Airplane%5C%202%20%3D%200)
This gives
![44800 - 3100m - 2500m = 0](https://tex.z-dn.net/?f=44800%20-%203100m%20-%202500m%20%3D%200)
![44800 - 5600m = 0](https://tex.z-dn.net/?f=44800%20-%205600m%20%3D%200)
Collect Like Terms
![5600m = 44800](https://tex.z-dn.net/?f=5600m%20%3D%2044800)
![m = 44800/5600](https://tex.z-dn.net/?f=m%20%3D%2044800%2F5600)
![m = 8\ min](https://tex.z-dn.net/?f=m%20%3D%208%5C%20min)
Substitute 8 for m in
![Airplane\ 1 = 44800 - 3100m](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%2044800%20-%203100m)
![Airplane\ 1 = 44800 - 3100 * 8](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%2044800%20-%203100%20%2A%208)
![Airplane\ 1 = 44800 - 24800](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%2044800%20-%2024800)
![Airplane\ 1 = 20000\ ft](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%2020000%5C%20ft)
Method 3:
We have that:
![Airplane\ 1 = 44800 - 3100m](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%2044800%20-%203100m)
![Airplane\ 2 = 2500m](https://tex.z-dn.net/?f=Airplane%5C%202%20%3D%202500m)
Since they are to be at the same altitude, then
The ratio of their altitudes must be 1
i.e.
![\frac{Airplane\ 1}{Airplane\ 2} = 1](https://tex.z-dn.net/?f=%5Cfrac%7BAirplane%5C%201%7D%7BAirplane%5C%202%7D%20%3D%201)
![\frac{44800 - 3100m}{2500m} = 1](https://tex.z-dn.net/?f=%5Cfrac%7B44800%20-%203100m%7D%7B2500m%7D%20%3D%201)
Cross Multiply
![44800 - 3100m = 1 * 2500m](https://tex.z-dn.net/?f=44800%20-%203100m%20%3D%201%20%2A%202500m)
![44800 - 3100m = 2500m](https://tex.z-dn.net/?f=44800%20-%203100m%20%3D%202500m)
Collect Like Terms
![44800 = 2500m + 3100m](https://tex.z-dn.net/?f=44800%20%3D%202500m%20%2B%203100m)
![44800 = 5600m](https://tex.z-dn.net/?f=44800%20%3D%205600m)
![5600m = 44800](https://tex.z-dn.net/?f=5600m%20%3D%2044800)
![m = 44800/5600](https://tex.z-dn.net/?f=m%20%3D%2044800%2F5600)
![m = 8\ min](https://tex.z-dn.net/?f=m%20%3D%208%5C%20min)
Substitute 8 for m in
![Airplane\ 1 = 44800 - 3100m](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%2044800%20-%203100m)
![Airplane\ 1 = 44800 - 3100 * 8](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%2044800%20-%203100%20%2A%208)
![Airplane\ 1 = 44800 - 24800](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%2044800%20-%2024800)
![Airplane\ 1 = 20000\ ft](https://tex.z-dn.net/?f=Airplane%5C%201%20%3D%2020000%5C%20ft)
Hence;
<em>Their altitudes must be 20000ft</em>
<em>Though the three methods applied uses the same logic at some point, the first method applied is still the easiest and it is a straight forward method that could be applied in solving the question.</em>
<em>Method 3 is the most difficult.</em>