1. **State the problem:** A kite is flying 100 ft above the ground and moving horizontally at 6 ft/s. We want to find the rate at which the angle $\theta$ between the string and the horizontal is decreasing when 200 ft of string is let out. 2. **Define variables:** - Let $x$ be the horizontal distance of the kite from the person (in ft). - Let $s$ be the length of the string (in ft). - Let $\theta$ be the angle between the string and the horizontal (in radians). 3. **Given:** - Height $h = 100$ ft (constant) - Horizontal speed $\frac{dx}{dt} = 6$ ft/s - String length $s = 200$ ft 4. **Relate variables using the Pythagorean theorem:** $$s^2 = x^2 + h^2$$ 5. **Differentiate both sides with respect to time $t$:** $$2s \frac{ds}{dt} = 2x \frac{dx}{dt} + 0$$ Since the string length is increasing as the kite moves away, $\frac{ds}{dt}$ is the rate at which the string length changes. 6. **Find $x$ when $s=200$ ft:** $$x = \sqrt{s^2 - h^2} = \sqrt{200^2 - 100^2} = \sqrt{40000 - 10000} = \sqrt{30000} = 100\sqrt{3}$$ 7. **Calculate $\frac{ds}{dt}$:** $$2 \times 200 \times \frac{ds}{dt} = 2 \times 100\sqrt{3} \times 6$$ $$400 \frac{ds}{dt} = 1200 \sqrt{3}$$ $$\frac{ds}{dt} = \frac{1200 \sqrt{3}}{400} = 3 \sqrt{3}$$ ft/s 8. **Express $\theta$ in terms of $x$ and $h$:** $$\tan \theta = \frac{h}{x}$$ 9. **Differentiate $\theta$ with respect to time:** $$\sec^2 \theta \frac{d\theta}{dt} = -\frac{h}{x^2} \frac{dx}{dt}$$ 10. **Calculate $\sec^2 \theta$:** Since $\tan \theta = \frac{h}{x}$, $$\sec^2 \theta = 1 + \tan^2 \theta = 1 + \left(\frac{h}{x}\right)^2 = 1 + \frac{10000}{(100\sqrt{3})^2} = 1 + \frac{10000}{30000} = 1 + \frac{1}{3} = \frac{4}{3}$$ 11. **Substitute values to find $\frac{d\theta}{dt}$:** $$\frac{4}{3} \frac{d\theta}{dt} = -\frac{100}{(100\sqrt{3})^2} \times 6 = -\frac{100}{30000} \times 6 = -\frac{6}{300} = -\frac{1}{50}$$ 12. **Solve for $\frac{d\theta}{dt}$:** $$\frac{d\theta}{dt} = -\frac{1}{50} \times \frac{3}{4} = -\frac{3}{200} = -0.015$$ rad/s 13. **Interpretation:** The negative sign indicates the angle is decreasing at a rate of $0.015$ radians per second. **Final answer:** The angle between the string and the horizontal is decreasing at a rate of $0.015$ rad/s when 200 ft of string have been let out.