1.
A rock is thrown into a still pond and causes a circular ripple. When the radius of the ripple is 15 inches, it is increasing at a rate of 3 inches per second.
How much does the circumference of the ripple increase over the next second?
Worksheet Related to Related Rates
Each scenario below describes how changing quantities are related to each other. Eventually, we will use derivatives to describe how the rates of change of these quantities are related.
But for now, let’s make sure that we understand how to describe the relation between the quantities. In each case, draw a picture, label quantities that are changing with variables, and write down one or more equations relating these changing quantities.
1.
2.
A street light is at the top of a 16 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 4 ft/sec along a straight path.
As she walks, the tip of her shadow moves along the ground. When she is 20 feet away from the pole, is she or the top of her shadow moving faster? By approximately how much?
3.
Suppose that two boats leave a dock at different times. One heads due north, the other due east. When the first boat is 33 miles from the dock it is traveling at a speed of 37 mph. At the same time, the second boat is 65 miles from the dock and is traveling at a speed of 26 mph.
If the boats maintain their same speed and heading for another hour, by how much will the distance between them change?
4.
A potter forms a piece of clay into a right circular cylinder. As she rolls it, the height \(h\) of the cylinder increases and the radius \(r\) decreases. Assume that no clay is lost in the process. When the radius is 3 cm and the height is 6 cm, the height is increasing by 0.6 cm/sec.
If this rate continues, what will the dimensions of the cylinder be 2 seconds later?
5.
Gravel is being dumped from a conveyor belt at a rate of \(40{\text{ft}}^{3}/\text{min}\text{.}\) It forms a pile in the shape of a right circular cone whose base diameter and height are always the same.
Which will take longer: to increase the height of the pile from 10 ft to 11 ft, or from 11 ft to 12 ft?
