# Thread: Related Rates

1. ## Related Rates

A kite 118 ft above the ground moves horizontally at a speed of 6ft/s. At what rate is the angle between the string and the horizontal decreasing when 236 ft of string have been let out?

2. We want $\displaystyle \frac{d{\theta}}{dt}$ when dx/dt=6.

The kite's string makes the hypoteneuse of the triangle. When 236 of string

is out then the angle is

$\displaystyle sin(\frac{118}{236})=sin(\frac{1}{2})=\frac{\pi}{6 }$

at that instant.

$\displaystyle x=118cot({\theta})$

$\displaystyle \frac{dx}{dt}=-118csc^{2}({\theta})\frac{d{\theta}}{dt}$

$\displaystyle \frac{d{\theta}}{dt}=\frac{-sin^{2}({\theta})}{118}\frac{dx}{dt}$

Now, using $\displaystyle {\theta}=\frac{\pi}{6} \;\ and \;\ \frac{dx}{dt}=6$

$\displaystyle \frac{d{\theta}}{dt}=\frac{-sin^{2}(\frac{\pi}{6})}{118}(6)=\frac{-3}{236} \;\ rad/sec$

3. Hello, asnxbbyx113!

A kite 118 ft above the ground moves horizontally at a speed of 6 ft/s.
At what rate is the angle between the string and the horizontal decreasing
when 236 ft of string have been let out?
Code:
                   x
* - - - - - - - - - - - *
|                 θ  *
|                 *
|              *
118 |           *
|        *
|     *
|  *
- * - - - - - - - - - - - - -

We have: .$\displaystyle \tan\theta\:=\:\frac{118}{x} \:=\:118x^{-1}$

Differentiate with respect to time: .$\displaystyle \sec^2\theta\left(\frac{d\theta}{dt}\right) \:=\:-118x^{-2}\left(\frac{dx}{dt}\right)$

. . And we have: .$\displaystyle \frac{d\theta}{dt}\;=\;-\frac{118\cos^2\theta}{x^2}\left(\frac{dx}{dt}\rig ht)$ [1]

When the string is $\displaystyle 236$ feet, Pythagorus gives us:
. . $\displaystyle x^2 + 118^2 \:=\:236^2\quad\Rightarrow\quad x^2\:=\:41,772\quad\Rightarrow\quad x \:=\:\sqrt{41,772}$

. . Then: .$\displaystyle \cos\theta \:=\:\frac{\sqrt{41,772}}{236}\quad\Rightarrow\qua d\cos^2\theta \:=\:\frac{41,772}{55,696}$

. . And we are told that: .$\displaystyle \frac{dx}{dt} \,=\,6$ ft/sec.

Substitute into [1]: .$\displaystyle \frac{d\theta}{dt} \;= \;-\frac{118(\frac{41,772}{55,696})}{41,772}(6) \;=\;-\frac{3}{236}$ radians/sec