1. ## Related Rates

A girl flies a kite at a height of 300 ft, the wind carrying the kite horizontally away from her at a rate of 25 ft/sec. How fast must she let out the string when the kite is 500 ft away from her?

i set up a triangle

y(t) = 300, y'(t)= 0

x(t) = 400 (since its a 3-4-5 triangle) x'(t) = 25

s(t) = 500 s'(t) = ? <--- hypotenuse

so the cos(theta) = 300/500 = .927295 radians

now, how can i get s'?

2. That's not the way to do it. As it is, you are analysing it at the condition when the lenght of the string is already 500 ft.
You must analyze it in the conditions before the length of the string is 500 ft.

Here it is assumed that the string is in a straight line, like the hypotenuse of a right triangle. [In reality, it is not--it sags a lot.]

----x at the top, horizontal leg.
----y=300 as the vertical leg
----z as the hypotenuse

So,
z = sqrt(x^2 +300^2)
z = sqrt(x^2 +90,000)

You are to find dz/dt when z = 500 ft.

z = sqrt(x^2 +90,000)
Differentiate both sides with respect to time t,
dz/dt = (1/2)(x^2 +90,000)^(-1/2) *2x *dx/dt
dz/dt = x (dx/dt) / sqrt(x^2 +90,000) --------------(i)

When z = 500ft, what is dz/dt?
You are solving for that.

When z = 500ft, what is dx/dt?
dx/dt is always 25 ft/sec anytime.

When z = 500ft, what is x?
z = sqrt(x^2 +90,000)
500 = sqrt(x^2 +90,000)
Square both sides,
250,000 = x^2 +90,000
x^2 = 250,000 -90,000
x^2 = 160,000
x = 400 ft

Hence, at z=500ft,
dz/dt = x (dx/dt) / sqrt(x^2 +90,000) --------------(i)
dz/dt = 400(25) / sqrt(400^2 +90,000)
dz/dt = 400(25) / sqrt(250,000)
dz/dt = 400(25) /500
dz/dt = 20 ft/sec

Therefore, she must let the string out at the rate of 20 feet per second at that time. -----------------answer.

3. Hello, mistykz!

A girl flies a kite at a height of 300 ft,
the wind carrying the kite horizontally away from her at a rate of 25 ft/sec.
How fast must she let out the string when the kite is 500 ft away from her?
I wouldn't introduce trigonometry . . .
Code:
      B        x        C
* - - - - - - - - * →
|              *
|           *
300 |        * S
|     *
|  *
A * - - - - - - - - - - - - -

The girl is at $\displaystyle A.$
The kite is at height of 300 feet: $\displaystyle AB = 300$
. . and is moving from point $\displaystyle C$ at the rate of $\displaystyle \frac{dx}{dt} = 25$ ft/sec.
The string has length: $\displaystyle S \,=\,AC$

Pythagorus says: .$\displaystyle S^2\:=\:x^2+300^2$ .[1]

Differentiate with respect to time: .$\displaystyle 2S\left(\frac{dS}{dt}\right) \:=\:2x\left(\frac{dx}{dt}\right)$

. . Hence, we have: .$\displaystyle \frac{dS}{dt} \:=\:\frac{x}{S}\cdot\frac{dx}{dt}$ .[2]

When $\displaystyle S = 500$, [1] becomes: .$\displaystyle 500^2 \:=\:x^2+300^2\quad\Rightarrow\quad x \:=\:400$

Substitute into [2]: .$\displaystyle \frac{dS}{dt} \:=\:\frac{400}{500}(25) \:=\:20$ ft/sec