Step-by-step guide to tackling related rates questions with changing angles

📚 Quick Study Guide

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• Trigonometric Ratios: Remember SOH CAH TOA! $\sin(\theta) = \frac{opposite}{hypotenuse}$, $\cos(\theta) = \frac{adjacent}{hypotenuse}$, $\tan(\theta) = \frac{opposite}{adjacent}$.
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• Chain Rule: If $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. This is crucial for related rates!
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• Implicit Differentiation: Differentiate each term with respect to time ($t$). Remember to apply the chain rule when differentiating trigonometric functions of $\theta$ with respect to $t$.
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• Angle Conventions: Ensure your calculator is in the correct mode (radians or degrees) depending on the problem. Radians are preferred in calculus.
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• Common Formulas: Pythagorean Theorem ($a^2 + b^2 = c^2$), area of a triangle ($A = \frac{1}{2}bh$), and trigonometric identities (e.g., $\sin^2(\theta) + \cos^2(\theta) = 1$).

🧪 Practice Quiz

1. A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the angle between the top of the ladder and the wall changing when the angle is $\frac{\pi}{3}$?
   1. $\frac{1}{5}$ rad/s
   2. $-\frac{1}{5}$ rad/s
   3. $\frac{1}{10}$ rad/s
   4. $-\frac{1}{10}$ rad/s


2. A kite is flying at a constant height of 150 ft. The kite is moving horizontally at a rate of 10 ft/s. At what rate is the angle between the string and the horizontal decreasing when 300 ft of string has been let out?
   1. $\frac{1}{3}$ rad/s
   2. $-\frac{1}{3}$ rad/s
   3. $\frac{1}{6}$ rad/s
   4. $-\frac{1}{6}$ rad/s


3. A spotlight on the ground shines on a wall 12 m away. A man 2 m tall walks from the spotlight toward the wall at a rate of 1.6 m/s. How fast is the angle of elevation of the top of the man's head changing when he is 4 m from the wall?
   1. 0.08 rad/s
   2. -0.08 rad/s
   3. 0.16 rad/s
   4. -0.16 rad/s


4. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the angle between the rope and the horizontal changing when 2 m of rope is out?
   1. $\frac{\sqrt{3}}{4}$ rad/s
   2. $-\frac{\sqrt{3}}{4}$ rad/s
   3. $\frac{\sqrt{3}}{2}$ rad/s
   4. $-\frac{\sqrt{3}}{2}$ rad/s


5. A plane is flying horizontally at an altitude of 1 mile and a speed of 500 mi/h. It passes directly over a radar station. At what rate is the angle between the horizontal and the line joining the radar station to the plane changing when the plane is 2 miles away from the station?
   1. -100 rad/hr
   2. 100 rad/hr
   3. 200 rad/hr
   4. -200 rad/hr


6. A security camera is mounted on a wall 9 ft above the floor. A man 6 ft tall is walking away from the wall at a rate of 2 ft/s. How fast is the angle of elevation of the top of the man's head changing when he is 15 ft away from the wall?
   1. -0.0024 rad/s
   2. 0.0024 rad/s
   3. 0.0012 rad/s
   4. -0.0012 rad/s


7. A lighthouse is located on a small island 3 km away from the nearest point $P$ on a straight shoreline and its light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from $P$?
   1. $30\pi$ km/min
   2. $60\pi$ km/min
   3. $96\pi$ km/min
   4. $48\pi$ km/min