A Kite 75 feet above the ground moves horizontally at a speed of 4 ft/s. At what rate is the angle between the string and the ground decreasing when 100 feet of string is out?

Question

A Kite 75 feet above the ground moves horizontally at a speed of  4 ft/s. At what rate is the angle between the string and the ground decreasing when 100 feet of string is out?

Answer 1

Draw the following figure

Consider the right triangle P K G , the right angle being at G.

P denotes for the person that flies the kite

K denotes the kite

G where the vertical line through the kite meets the ground.

KG = h =75 ft

PK = s = 100 ft

PG =x

and using the Pythagorean Theorem

x² + h² = s²

Let φ = ∠ KPG. Then tan φ = ( h) / x , where x is the horizontal distance between P and G.

Since tan φ = ( h) / x the φ = tan^-1 (h/ x) We are asked to find dφ / d t.

dφ / d t. = ( d/ d t) [ tan^-1 (h/ x) ]

dφ / d t.=[ x²/ ( x² +h² ) [ d /d x( h/ x)]

dφ / d t.=[ x²/ ( x² +h² ) [ d /d x( h/ x)] [ d x / d t ]

dφ / d t.=[ x²/ ( x² +h² ) [ - h/ x² ] [ d x / d t ]

dφ / d t.=[ - h/ ( s² ) [ d x / d t ]

dφ / d t.= [- 75 / 10000 ] [ 4 ] rad / sec

dφ / d t. = - 3/ 100 rad / sec