Question

A kite 100 ft above the ground moves horizontally at a speed of 7 ft/s. At what rate is the angle (in radians) between the string and the horizontal decreasing when 200 ft of string have been let out?

Answer 1

Answer:0.07 rad/s

Explanation:

Given

kite is 100 ft high

and moves horizontally at 7 ft/s

Total string let out =200 ft

String length(l), vertical(y) & Horizontal(x) distance of kite will form a right angle triangle

$L^2=y^2+x^2$

Differentiating both side

$0=2y\frac{\mathrm{d} y}{\mathrm{d} t}+2x\frac{\mathrm{d} x}{\mathrm{d} t}$

$y\frac{\mathrm{d} y}{\mathrm{d} t}=-x\frac{\mathrm{d} x}{\mathrm{d} t}$

$100\times \frac{\mathrm{d} y}{\mathrm{d} t}=-\sqrt{3}\times 100\times \frac{\mathrm{d} x}{\mathrm{d} t}$

$\frac{\mathrm{d} y}{\mathrm{d} t}=7\sqrt{3}$

Now $Lcos\theta =x$

Differentiating

$Lsin\theta \times \frac{\mathrm{d} \theta }{\mathrm{d} t}=\frac{\mathrm{d} x}{\mathrm{d} t}$

$200\times \frac{100}{200}\times \frac{\mathrm{d} \theta }{\mathrm{d} t}=7$

$\frac{\mathrm{d} \theta }{\mathrm{d} t}=0.07 rad/s$