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? (Enter your answer as a fraction.)

Answer 1

The rate at which the angle between the string and the horizontal is decreasing is 0 radians per second when 200 ft of string have been let out.

Step-by-step explanation:

To find the rate at which the angle between the string and the horizontal is decreasing, we can use trigonometry and related rates.

Let x be the horizontal distance from the kite to the person letting out the string, and let y be the height of the kite above the ground. We are given that y is constant at 100 ft and dx/dt = 7 ft/s. We want to find dθ/dt when x = 200 ft.

Using the Pythagorean theorem, we can find the length of the string (L) when x = 200 ft: L^2 = x^2 + y^2. Substituting the given values, we have L = √(200^2 + 100^2) = √50000 = 100√5 ft.

The angle (θ) between the string and the horizontal can be found using inverse trigonometry: sin(θ) = y / L = 100 / (100√5) = 1 / √5. Taking the inverse sine of both sides, we have θ = arcsin(1/√5).

Differentiating both sides of the equation with respect to time (t), we get dθ/dt = d(arcsin(1/√5)) / dt.

Since arcsin(1/√5) is a constant, its derivative with respect to t is 0. Therefore, dθ/dt = 0.