26-27 Time Rates: Kite moving horizontally

$s^2 = x^2 + 40^2$

$2s \dfrac{ds}{dt} = 2x \dfrac{dx}{dt}$

$s \dfrac{ds}{dt} = x \dfrac{dx}{dt}$
 

when s = 50 ft
50² = x² + 40²
x = 30 ft

 

Thus,
$50 \dfrac{ds}{dt} = 30(\frac{22}{3})$

$\dfrac{ds}{dt} = 4.4 \, \text{ ft/sec}$           answer

Slope
$m = \dfrac{40}{x}$

$\dfrac{dm}{dt} = \dfrac{-40}{x^2}\,\dfrac{dx}{dt}$
 

From Solution 26, x = 30 ft when s = 50 ft
$\dfrac{dm}{dt} = \dfrac{-40}{30^2}\,(\frac{22}{3})$

$\dfrac{dm}{dt} = -\frac{44}{135} \, \text{ rad/sec}$           answer