Calculus Related Rates Test Questions: Ladders, Cones, and Spheres

📚 Quick Study Guide

📐 Related Rates Basics: These problems involve finding the rate at which a quantity is changing by relating it to other quantities whose rates of change are known. Implicit differentiation is key! 💡 General Steps:
• 1. Draw a diagram.
• 2. Identify known rates and the rate you want to find.
• 3. Write an equation relating the variables.
• 4. Differentiate both sides with respect to time ($t$).
• 5. Substitute known values and solve for the unknown rate.
🪜 Ladder Problems: Often involve the Pythagorean theorem ($a^2 + b^2 = c^2$). Remember that the length of the ladder ($c$) is constant. 🍦 Cone Problems: Usually involve similar triangles to relate the radius ($r$) and height ($h$) of the cone. The volume of a cone is $V = \frac{1}{3}\pi r^2 h$. ⚽ Sphere Problems: The volume of a sphere is $V = \frac{4}{3}\pi r^3$ and the surface area is $A = 4\pi r^2$.

Practice Quiz

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 of the ladder is 5 feet from the wall?
   1. $\frac{5}{6}$ ft/sec
   2. $\frac{5}{12}$ ft/sec
   3. $-\frac{5}{6}$ ft/sec
   4. $-\frac{5}{12}$ ft/sec
2. A conical tank (vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate at which the depth of the water is increasing when the water is 8 feet deep.
   1. $\frac{9}{16\pi}$ ft/min
   2. $\frac{5}{36\pi}$ ft/min
   3. $\frac{5}{16\pi}$ ft/min
   4. $\frac{9}{36\pi}$ ft/min
3. A spherical balloon is being inflated. If the radius is increasing at a rate of 4 cm/sec, find the rate at which the volume is increasing when the radius is 8 cm.
   1. $512\pi$ cm$^3$/sec
   2. $1024\pi$ cm$^3$/sec
   3. $2048\pi$ cm$^3$/sec
   4. $256\pi$ cm$^3$/sec
4. 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. $\frac{3}{4}$ ft/s
   2. $-\frac{3}{4}$ ft/s
   3. $\frac{4}{3}$ ft/s
   4. $-\frac{4}{3}$ ft/s
5. Water is leaking out of an inverted conical tank at a rate of 10000 cm$^3$/min at the same time that water is being pumped into the tank at a rate of 12000 cm$^3$/min. The tank has height 6 m and the diameter at the top is 4 m. If the water level is 5 m, find the rate at which the water level is rising.
   1. $\frac{1}{50\pi}$ m/min
   2. $\frac{1}{25\pi}$ m/min
   3. $\frac{2}{25\pi}$ m/min
   4. $\frac{3}{50\pi}$ m/min
6. 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. $\frac{8}{125}$ rad/s
   2. $\frac{4}{125}$ rad/s
   3. $\frac{2}{125}$ rad/s
   4. $\frac{6}{125}$ rad/s
7. A point P is moving along the curve whose equation is $y = \sqrt{x}$. If x is increasing at the rate of 4 units/second, when x = 4, what is the rate of change of the distance between P and the point (2,0)?
   1. $\frac{2}{\sqrt{5}}$ units/sec
   2. $\frac{3}{\sqrt{5}}$ units/sec
   3. $\frac{4}{\sqrt{5}}$ units/sec
   4. $\frac{5}{\sqrt{5}}$ units/sec