To solve this problem, we will first start by finding the initial angle between the kite string and the ground (horizontal). Since we are given the height of the kite above the ground and the length of the string let out, we can think of this as a right triangle. The string of the kite would be the hypotenuse, and the height from the ground the kite is flown will be one of the sides of the triangle.
We'll use the concept of trigonometry to find the initial angle. The cosine (cos) of an angle in a right triangle is equal to the adjacent side (height) divided by the hypotenuse (length of the string). We're given that the height of the kite from the ground is 50ft and the length of the string let out is 200ft. So, cos(initial angle) = 50/200 = 0.25.
Our next step will be to find the rate at which this angle is decreasing. For this, we need to apply derivatives and the chain rule. The derivative of cosine (cos) with respect to the angle is negative sine (-sin). We also know that the speed of the kite is equal to the distance (horizontal in this case) over time. Hence, the rate of change of the distance (horizontal distance) with respect to time 't' would be the speed of the kite. Therefore, the rate of change of cos(angle) with respect to time 't' would be equal to -sin(angle) times the horizontal speed.
To implement this, we have to first find sin(initial angle). We can determine this value using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (string length) is equal to the sum of the squares of the lengths of the other sides (height and horizontal distance). We use this theorem to find the horizontal distance which comes out to be √(200^2 - 50^2) ft. So, sin(initial angle) can be computed as the horizontal distance divided by the length of the string, which equals 0.9682458365518543.
Multiplying this by the speed of the kite (-4 ft/sec because it's decreasing), we get the rate at which the angle is decreasing as -3.872983346207417 radians per second.
So, when the kite is 50ft above the ground and 200ft of string has been let out, the angle between the string and the horizontal is decreasing at a rate of approximately -3.873 radians per second.