Real-World Examples of Related Rates in Motion

📚 Quick Study Guide

• 📐 Related rates problems involve finding the rate at which a quantity changes by relating it to other quantities whose rates of change are known.
• 📝 The chain rule is fundamental: $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$.
• 💡 Common formulas used: Area of a circle ($A = \pi r^2$), Volume of a sphere ($V = \frac{4}{3} \pi r^3$), Pythagorean theorem ($a^2 + b^2 = c^2$).
• 🧭 Steps to solve: (1) Draw a diagram, (2) Identify knowns and unknowns, (3) Write an equation relating the variables, (4) Differentiate with respect to time, (5) Substitute known values and solve.
• ⏱️ Units are crucial! Ensure consistent units throughout the problem. For example, if distance is in meters, and time is in seconds, then the rate is in meters/second.

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. 0.6 ft/s
   2. 0.75 ft/s
   3. 0.8 ft/s
   4. 1.25 ft/s
2. A spherical balloon is inflated so that its radius increases at a rate of 2 cm/s. How fast is the volume of the balloon increasing when the radius is 5 cm?
   1. $100\pi$ cm³/s
   2. $200\pi$ cm³/s
   3. $300\pi$ cm³/s
   4. $400\pi$ cm³/s
3. A conical tank (with vertex down) is 10 ft across the top and 12 ft deep. If water is flowing into the tank at a rate of 10 ft³/min, find the rate at which the depth of the water is increasing when the water is 8 ft deep.
   1. $\frac{9}{16\pi}$ ft/min
   2. $\frac{5}{16\pi}$ ft/min
   3. $\frac{2}{9\pi}$ ft/min
   4. $\frac{3}{8\pi}$ ft/min
4. A man walks along a straight path at a speed of 4 ft/s. A searchlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 ft from the point on the path closest to the searchlight?
   1. 0.08 rad/s
   2. 0.10 rad/s
   3. 0.12 rad/s
   4. 0.16 rad/s
5. 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. $\frac{8}{\sqrt{65}}$ m/s
   2. $\frac{7}{\sqrt{65}}$ m/s
   3. $\frac{9}{\sqrt{65}}$ m/s
   4. $\frac{6}{\sqrt{65}}$ m/s
6. Oil is leaking from a tanker and spreads in a circular pattern. If the radius of the oil slick is increasing at a constant rate of 1.5 km/h, how fast is the area of the slick increasing when the radius is 3 km?
   1. $4.5\pi$ km²/h
   2. $6\pi$ km²/h
   3. $9\pi$ km²/h
   4. $12\pi$ km²/h
7. A kite is 100 ft above the ground and is moving horizontally away from the person holding the string at a rate of 8 ft/s. At what rate is the angle between the string and the vertical changing when 200 ft of string has been let out?
   1. 0.03 rad/s
   2. 0.04 rad/s
   3. 0.05 rad/s
   4. 0.06 rad/s