Question

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?

Answer 1

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

Answer 2

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

Answer 3

your*

Answer 4

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

Answer 5

let me know if youre still here

Answer 6

yeah i drew one

Answer 7

sorry, you still there?

Answer 8

still here!

Answer 9

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?

Answer 10

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

Answer 11

triangle sounds good, what kind of triangle

Answer 12

right

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

i was using x and y for that

Answer 18

oh whoops!

Answer 19

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

Answer 20

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

Answer 21

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

Answer 22

do you know the pythagorean theorem?

Answer 23

a^2 + b^2 = c^2

Answer 24

so what is c here in our problem

Answer 25

we don't know c?

Answer 26

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

Answer 27

what does c stand for though

Answer 28

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

Answer 29

the distance between planes

Answer 30

right

Answer 31

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

Answer 32

?

Answer 33

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

Answer 34

dt/dt = y' ?

Answer 35

192mi/hr. i never would have gotten that

Answer 36

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

Answer 37

was I right is what I want to know :)

Answer 38

haha i have to find that out....