# Related Rates

• Jan 7th 2007, 11:43 AM
asnxbbyx113
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?
• Jan 7th 2007, 12:42 PM
galactus
We want $\frac{d{\theta}}{dt}$ when dx/dt=6.

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

is out then the angle is

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

at that instant.

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

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

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

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

$\frac{d{\theta}}{dt}=\frac{-sin^{2}(\frac{\pi}{6})}{118}(6)=\frac{-3}{236} \;\ rad/sec$
• Jan 7th 2007, 01:08 PM
Soroban
Hello, asnxbbyx113!

Quote:

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

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

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

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

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

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

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