AB Calculus - Hardtke Assignment 4.1: Related Rates A Name ___________________________________

Due Date: Monday, 11/18 (4 pts) Preliminary: implicit differentiation skill for Related Rates Problems:

A.

B.

C.

D.

Example 1: Water is flowing into a cone of height = 16 cm and r = 4 cm) at 2

⁄ .

How fast is the water level rising when it is 5, 10, 15 cm deep? What rate is given? Now substitute data to find rate when h = 5: What rate is to be found? (Static) Model for V of a cone: Now substitute data to find rate when h = 10: Simplify the number of variables by converting r in terms of h: Now substitute data to find rate when h = 15: (You can’t always do this, but use it when possible)

Differentiate implicitly w/ respect to time Now you have your RELATED RATES MODEL: Is the rate at which the level is rising changing as you expected? Always use this to check accuracy.

Example 2: If the bottom of a 6m ladder resting against a wall is pulled away at a rate of

⁄ , how fast will the top of

the ladder be falling when it reaches a height of 3m? Rate(s) given: Rate to be found: Static model:

RELATED RATES MODEL:

Specific case prediction: (Sometimes helpful to draw new diagram for each case)

Over

1. Go to the assignment page of our class website and watch Ms. Hardtke’s video: Inflating a Balloon

2. If air is pumped into a spherical balloon so its volume increases at a rate of 100

⁄ How fast is the radius of the balloon increasing when the diameter is 30 cm?

3. 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 6ft from the wall?

4. At noon, ship A is 150km west of ship B. Ship A is sailing east at 35 ⁄ and ship B is sailing north at 25 ⁄ . How

fast is the distance between the ships changing at 4:00pm?

AB Calculus - Hardtke Assignment 4.1: Related Rates SOLUTION KEY

2. If air is pumped into a spherical balloon so its volume increases at a rate of 100

⁄ How fast is the radius of the balloon increasing when the diameter is 30 cm?

3. 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 6ft from the wall?

4. A noon, ship A is 150km west of ship B. Ship A is sailing east at 35km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00pm?

(12 from 4.1)