To solve this problem, let’s define the key variables:

1. Let $h = 100$ feet (the height of the kite above the ground).
2. The kite is moving horizontally at a speed of $\frac{dx}{dt} = 4$ ft/s.
3. Let $s$ be the length of the string let out, which is given as 150 ft.
4. Let $\theta$ be the angle between the string and the horizontal line.

We are asked to find the rate at which this angle $\theta$ is decreasing. This corresponds to finding $\frac{d\theta}{dt}$ when $s = 150$ ft.

Step 1: Set Up the Relationship

Using trigonometry, we know: $\cos \theta = \frac{x}{s}$ where $x$ is the horizontal distance from the person holding the kite to the point directly under the kite. Also, since $x$ and $h$ form a right triangle with hypotenuse $s$, we have: $s^2 = x^2 + h^2$

Step 2: Differentiate with Respect to Time

Differentiate both sides of $s^2 = x^2 + h^2$ with respect to $t$: $2s \frac{ds}{dt} = 2x \frac{dx}{dt}$ Since the length of the string is constant ($s = 150$ ft), $\frac{ds}{dt} = 0$. Simplifying, we get: $0 = 2x \frac{dx}{dt}$ So