Avoiding Pitfalls in Related Rates Calculus: Ladder & Cone Examples

📚 Quick Study Guide

🔍 General Strategy: Identify rates, known variables, and what you need to find. Draw a diagram! Write an equation relating the variables. Differentiate with respect to time ($t$). Substitute known values and solve for the unknown rate. 📐 Ladder Problems: Usually involve the Pythagorean theorem: $a^2 + b^2 = c^2$, where $a$ and $b$ are the distances from the wall and ground, respectively, and $c$ is the length of the ladder (constant). 🍦 Cone Problems: Typically involve similar triangles and volume formulas. The volume of a cone is $V = \frac{1}{3}\pi r^2 h$. If the ratio of $r$ to $h$ is constant, express $r$ in terms of $h$ (or vice versa) before differentiating to reduce variables. 💡 Implicit Differentiation: Remember the chain rule! When differentiating a variable with respect to time, multiply by $\frac{d(variable)}{dt}$. For example, $\frac{d}{dt}(x^2) = 2x \frac{dx}{dt}$. ⏱️ Units: Always include units in your final answer. Pay attention to the units given in the problem! ➕ Positive/Negative Rates: A positive rate indicates an increasing quantity, while a negative rate indicates a decreasing quantity.

🧪 Practice Quiz

1. Question 1: A 13-foot ladder is leaning against a wall. If the foot of the ladder is pulled away from the wall at a rate of 2 ft/sec, how fast is the top of the ladder sliding down the wall when the foot is 5 feet from the wall?
   1. A) -1.92 ft/sec
   2. B) 1.92 ft/sec
   3. C) -2.00 ft/sec
   4. D) 2.00 ft/sec
2. Question 2: Water is leaking out of an inverted conical tank at a rate of 10000 cm³/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, find the rate at which water is being pumped into the tank.
   1. A) 40000$\pi$ cm³/min
   2. B) 50000$\pi$ cm³/min
   3. C) (10000 + 20000$\pi$) cm³/min
   4. D) (10000 + 50000$\pi$) cm³/min
3. Question 3: A point is moving along the curve $y = \sqrt{x}$. If $\frac{dx}{dt} = 3$ cm/sec, find $\frac{dy}{dt}$ when $x = 4$.
   1. A) $\frac{3}{2}$ cm/sec
   2. B) $\frac{3}{4}$ cm/sec
   3. C) $\frac{2}{3}$ cm/sec
   4. D) $\frac{4}{3}$ cm/sec
4. Question 4: A kite is 100 ft above the ground and moving horizontally at a rate of 8 ft/sec. At what rate is the angle between the string and the horizontal decreasing when 300 ft of string has been let out?
   1. A) 0.025 rad/sec
   2. B) 0.0267 rad/sec
   3. C) -0.0267 rad/sec
   4. D) -0.025 rad/sec
5. Question 5: Gravel is being dumped from a conveyor belt at a rate of 30 ft³/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?
   1. A) $\frac{3}{5\pi}$ ft/min
   2. B) $\frac{5\pi}{3}$ ft/min
   3. C) $\frac{3}{2\pi}$ ft/min
   4. D) $\frac{2\pi}{3}$ ft/min
6. Question 6: A man walks along a straight path at a speed of 4 ft/sec. 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. A) 0.16 rad/sec
   2. B) 0.096 rad/sec
   3. C) 0.08 rad/sec
   4. D) 0.12 rad/sec
7. Question 7: A spherical balloon is being inflated. If the radius is increasing at a rate of 2 cm/sec, find the rate at which the surface area is increasing when the radius is 5 cm. (Surface area of a sphere: $A = 4\pi r^2$)
   1. A) 20$\pi$ cm²/sec
   2. B) 40$\pi$ cm²/sec
   3. C) 80$\pi$ cm²/sec
   4. D) 160$\pi$ cm²/sec