Calculus Test Prep: Related Rates & Similar Triangles Questions

📚 Quick Study Guide

📐 Similar Triangles: If two triangles are similar, their corresponding sides are proportional. This means if $\triangle ABC \sim \triangle DEF$, then $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$. ⏰ Related Rates: Related rates problems involve finding the rate at which a quantity changes by relating it to other quantities whose rates of change are known. Key steps:
• ✍️ Write an equation relating the variables.
• 🔄 Differentiate both sides with respect to time ($t$).
• ➕ Substitute known values and solve for the unknown rate.
💡 Common Formulas:
• 📏 Area of a Triangle: $A = \frac{1}{2}bh$
• ➗ Pythagorean Theorem: $a^2 + b^2 = c^2$
• ➿ Volume of a Cone: $V = \frac{1}{3}\pi r^2 h$
🧭 Tips for Success:
• ✏️ Draw a diagram whenever possible.
• 📝 Clearly label all variables and rates.
• ✅ Double-check your units.

🧪 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. 2.08 ft/sec
   2. -0.96 ft/sec
   3. -2.08 ft/sec
   4. 0.96 ft/sec


2. A conical tank (with 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. $\frac{5}{16\pi}$ ft/min
   3. $\frac{16\pi}{5}$ ft/min
   4. $\frac{16\pi}{9}$ ft/min


3. A man walks along a straight path at a speed of 4 ft/sec. A searchlight is located on the ground 20 feet from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 feet from the point on the path closest to the searchlight?
   1. 0.08 rad/sec
   2. 0.16 rad/sec
   3. 0.24 rad/sec
   4. 0.32 rad/sec


4. A water tank has the shape of an inverted cone with a height of 10 meters and a radius of 4 meters at the top. If water is being pumped into the tank at a rate of 2 $m^3$/min, find the rate at which the water level is rising when the water is 5 meters deep.
   1. $\frac{1}{2\pi}$ m/min
   2. $\frac{5}{4\pi}$ m/min
   3. $\frac{2}{5\pi}$ m/min
   4. $\frac{1}{4\pi}$ m/min


5. A kite is flying at a height of 40 ft, and the wind is carrying it horizontally at a rate of 3 ft/sec. How fast must the string be let out when the kite is 50 ft away from the person holding the string?
   1. 1.8 ft/sec
   2. 2.4 ft/sec
   3. 3.6 ft/sec
   4. 4.8 ft/sec


6. A spotlight on the ground shines on a wall 12 m away. If 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. -0.8 m/s
   2. -0.4 m/s
   3. 0.4 m/s
   4. 0.8 m/s


7. An airplane is flying at an altitude of 5 miles toward a radar antenna. If the distance $s$ between the plane and the antenna is decreasing at a rate of 400 miles per hour when $s = 13$ miles, what is the horizontal speed of the plane?
   1. 462 mph
   2. 433 mph
   3. 369 mph
   4. 338 mph