ALKJFSALJF. Two small planes approach an airport, one flying due west at 120mi/hr and the other flying due north at 150mi/hr. Assuming they fly at the same constant elevation, how fast is the distance between the planes changing when the westbound plane is 180mi from the airport and the northbound plane is 225mi from the air port?

- anonymous

- chestercat

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- anonymous

hey, you keep posting problems i can't catch up, which do you want to work on?

- anonymous

haha sorry! i'm flustered. whatever you're heart desires to work on is good enough for me

- anonymous

your*

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## More answers

- anonymous

with these types of problems i always start by drawing the picture, did you do that?

- anonymous

let me know if youre still here

- anonymous

yeah i drew one

- anonymous

sorry, you still there?

- anonymous

still here!

- anonymous

ok so what does your picture look like? the question is asking for the rate of change of the distance. so we need an equation for the distance and take the derivative. how can we get an equation for the distance?

- anonymous

i kind of made a triangle between the two planes. i'm use to doing rate of change of distance with a point and equation of a line given

- anonymous

triangle sounds good, what kind of triangle

- anonymous

right

- anonymous

yes, so we are given rate of change of the sides of the triangle, so we want an equation for the distance as a function of the sides of the triangle

- anonymous

basically we want d = (something with x's and y's), because we know x,y, dx/dt, dy/dt, so if we take the derivative we can find dd/dt

- anonymous

say a is 180x and b is 225y then d(b-a)/dt=(180)x+(225)y
just a guess

- anonymous

hmm, now a and b is what you are using for the sides of the triangle?

- anonymous

i was using x and y for that

- anonymous

oh whoops!

- anonymous

so if you want you can take everything i said and substitute a for x and b for y, it's the same thing

- anonymous

now don't confuse 180,225 and the variables. it's 180 at a certain time, and x (or a), when you don't know what it is

- anonymous

what's a formula for d in terms of a and b

- anonymous

do you know the pythagorean theorem?

- anonymous

a^2 + b^2 = c^2

- anonymous

so what is c here in our problem

- anonymous

we don't know c?

- anonymous

wait 225^2 + 180^2 =83025 then take the square of that?

- anonymous

what does c stand for though

- anonymous

yes that is what c is when a and b are those numbers

- anonymous

the distance between planes

- anonymous

right

- anonymous

so what's the equation for the distance between the planes in terms of a and b

- anonymous

?

- amistre64

the distance formula is a^2 + b^2 = c^2 and we want to derive with respect to time:
(d/dt)(x^2) + (d/dt)(y^2) = (d/dt)(c^2)
(dx/dt)(2x) + (dy/dt)(2y) = (dc/dt)(2c)
To clean this up: (dx/dt) = x'; (dt/dt) = y'; and (dc/dt) = c'.
x'2x + y'2y = c'2c
solve for c':
x'2x + y'2y
----------- = c'
2c
the twos cancel to give us:
x'(x) + y'(y)
--------- = c'
c
y' = 150; x'=120; and c = whatever you got with the pythag. theorum.
x and y values are their current values of x=180; y=225
So lets plug those in:
120(180) + 150(225)
------------------- = c'
288.14

- anonymous

dt/dt = y' ?

- anonymous

192mi/hr. i never would have gotten that

- amistre64

lol....(dy/dt) = y' :)

- amistre64

was I right is what I want to know :)

- anonymous

haha i have to find that out....

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