Solved examples of related rates problems with angles and trigonometry

📚 Quick Study Guide

• 📐 Trigonometric Functions: Remember your basic trig functions: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.
• 🔄 Related Rates: These problems involve finding the rate at which a quantity changes by relating it to other quantities whose rates of change are known.
• 🔗 Chain Rule: Use the chain rule when differentiating composite functions. For example, if $y = \sin(\theta)$ and $\theta = f(t)$, then $\frac{dy}{dt} = \frac{dy}{d\theta} \cdot \frac{d\theta}{dt}$.
• 🧭 Angle Measures: Make sure your calculator is in the correct mode (degrees or radians) depending on the problem. Radians are often preferred in calculus.
• 📝 Implicit Differentiation: Differentiate equations with respect to time ($t$) even if the variable isn't explicitly a function of $t$.
• 💡 Problem-Solving Strategy:
  1. Read the problem carefully.
  2. Draw a diagram if possible.
  3. Identify known rates and the rate you want to find.
  4. Write an equation relating the variables.
  5. Differentiate both sides of the equation with respect to time.
  6. Substitute known values and solve for the unknown rate.

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 top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?
   1. A) -0.6 ft/s
   2. B) -0.75 ft/s
   3. C) -0.8 ft/s
   4. D) -1.33 ft/s
2. A kite is flying at a height of 40 ft and is moving horizontally away from the person holding the string at a rate of 5 ft/s. At what rate is the angle between the string and the horizontal decreasing when 80 ft of string has been let out?
   1. A) 0.04 rad/s
   2. B) 0.05 rad/s
   3. C) 0.025 rad/s
   4. D) 0.0125 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 building at a speed of 1.6 m/s. How fast is the length of his shadow on the building decreasing when he is 4 m from the building?
   1. A) 0.4 m/s
   2. B) 0.8 m/s
   3. C) 0.6 m/s
   4. D) 1.2 m/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 boat approaching the dock when it is 8 m from the dock?
   1. A) -1.008 m/s
   2. B) -0.992 m/s
   3. C) -0.984 m/s
   4. D) -0.976 m/s
5. A camera is located 5000 ft from a rocket launching pad. If the rocket is rising vertically at a rate of 880 ft/s when it is 4000 ft high, at what rate must the camera's angle of elevation change at that instant to keep the camera trained on the rocket?
   1. A) 0.105 rad/s
   2. B) 0.115 rad/s
   3. C) 0.125 rad/s
   4. D) 0.135 rad/s
6. A revolving beacon light is located 3 miles from a straight shoreline. If the light revolves at 4 revolutions per minute, how fast is the beam of light sweeping along the shoreline when it is 5 miles from the point on the shoreline nearest the beacon?
   1. A) 125.66 miles/min
   2. B) 135.66 miles/min
   3. C) 145.66 miles/min
   4. D) 155.66 miles/min
7. A hot air balloon rising straight up is tracked by an observer who is 2 km away from the launch point. At a certain moment, the angle of elevation is $\frac{\pi}{4}$, and it is changing at a rate of 0.2 rad/min. What is the speed of the balloon at that moment?
   1. A) 0.4 km/min
   2. B) 0.8 km/min
   3. C) 1.2 km/min
   4. D) 1.6 km/min