3. Applications of Derivatives

Related Rates — Quiz

Test your understanding of related rates with 5 practice questions.

Practice Questions

Question 1

A boat is pulled toward a dock by a rope attached to the bow of the boat and passing through a pulley on the dock $1 \text{ m}$ higher than the bow. If the rope is pulled in at a rate of $1 \text{ m/s}$, how fast is the boat approaching the dock when $10 \text{ m}$ of rope are out?

Question 2

A street light is mounted at the top of a $15 \text{ ft}$ tall pole. A man $6 \text{ ft}$ tall walks away from the pole at a speed of $5 \text{ ft/s}$ along a straight path. How fast is the tip of his shadow moving when he is $10 \text{ ft}$ from the base of the pole?

Question 3

Sand is being poured into a conical pile at a rate of $12 \pi \text{ ft}^3/\text{min}$. The height of the pile is always equal to its radius. How fast is the height of the pile increasing when the height is $6 \text{ ft}$?

Question 4

A balloon is rising vertically above a point A on the ground at a rate of $15 \text{ ft/s}$. A point B on the ground is $30 \text{ ft}$ from A. At what rate is the distance between the balloon and point B changing when the balloon is $40 \text{ ft}$ above point A?

Question 5

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