SOLUTION: Hello, I need help setting up this related rates question in order to find the starting equation: an airplane passes over an airport at noon traveling 500 miles per hour due West,

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Question 1022781: Hello, I need help setting up this related rates question in order to find the starting equation:
an airplane passes over an airport at noon traveling 500 miles per hour due West, and at 1 p.m. another airplane passes over the same airport at the same elevation traveling due north at 550 miles per hour. Assuming both planes maintain their equal elevations, how fast is the distance between them changing at 2:30 p.m.?
I think the use of the Pythagorean theorem is helpful, something like a(t)^2 +b(t)^2=c(t)^2, variable with repsect to t being a=500 and b=550. Am I doing this right? Thanks!
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Yes,
a%5E2%2Bb%5E2=c%5E2
Use implicit differentiation with respect to time,
2a%28da%2Fdt%29%2B2b%28db%2Fdt%29=2c%28dc%2Fdt%29
With
da%2Fdt=500
db%2Fdt=550
At 2:30, the first plane traveled,
a=%285%2F2%29%28500%29=1250
The second plane traveled,
b=%283%2F2%29%28550%29=825
We can calculate c using the Pythagorean theorem,
c%5E2=%281250%29%5E2%2B%28825%29%5E2
c%5E2=1562500%2B680625
c%5E2=2243125
c=25sqrt%283589%29
So then,
2%281250%29%28500%29%2B2%28825%29%28550%29=2%2825sqrt%283589%29%29%28dc%2Fdt%29
1250000%2B907500=50sqrt%283589%29%28dc%2Fdt%29
dc%2Fdt=%282157500%29%2F%2850sqrt%283589%29%29
dc%2Fdt=43150%2Fsqrt%283589%29
dc%2Fdt=%2843150%2F3589%29%2Asqrt%283589%29mph