Final answer:
The rate at which the distance between the man and the kite is changing, when the kite is 130 m away, is calculated using the Pythagorean theorem and is found to be 3.1 m/sec.
Step-by-step explanation:
Calculating Rate of Change in Distance
Given that a kite flies horizontally away at a constant speed of 8 m/sec and its height above the ground remains constant at 120 m, we can represent the situation with a right triangle. The vertical height from the man to the kite is one leg (120 m), and the horizontal distance from the man to the point directly below the kite will be varying as time passes. At the moment when the kite is 130 m away from the man, we use the Pythagorean theorem to find the hypotenuse (the direct distance from the man to the kite). Let's denote the horizontal distance by x and the rate at which this distance changes by dx/dt = 8 m/sec, the direct distance by r, and the rate at which r changes by dr/dt.
We have:
Height (h) = 120 m (constant)
Horizontal distance (x) = ??? varies over time
Direct distance (r) = 130 m
Using the Pythagorean theorem:
r^2 = h^2 + x^2
130^2 = 120^2 + x^2
x^2 = 130^2 - 120^2
x = 50
Since x changes over time, we differentiate both sides of r^2 = h^2 + x^2:
2r x dr/dt = 2x x dx/dt
2 x 130 x dr/dt = 2 x 50 x 8
dr/dt = (50 x 8)/130
So, the rate at which the distance between the man and the kite is changing is 3.1 m/sec.