Review Implicit Differentiation Take the following derivative

AP Calculus Unit 4 Day 10 Related Rates

Arrival Questions (PUT in YOUR NOTES) Assume that the width a and length b of a rectangle are functions of t. Write the expression for the area of the rectangle. A= ________ Take the derivative of the area expression with respect to t.

Arrival Questions (PUT in YOUR NOTES) Assume that the width a and length b of a rectangle are functions of t. Take the derivative of the area expression with respect to t.

Vocab Reminder THIS MEANS: 1) Rate of change of Area 2) Change in Area over time

Related Rates BASIC strategy (some variation based on problem) Diagram Rates Equation Derivative Substitute

More Examples – Add to your notes! Stacey is rolling out pizza dough into a circle when she discovers that the radius is increasing at the rate of 2 cm/sec. How fast is the circumference increasing when the radius is 10 cm?

And another… A circular piece of ice is melting. The area is decreasing by 4 ft 2 /min. How fast is the circumference changing when the area is exactly feet?

NOTES Keys to related rates: 1) Set up an equation that relates all your variables. 1a) You may have to set up two equations and substitute one into the other. 1b) You can fill in anything you KNOW remains constant over time. 2) Take the derivative with respect to time (“t”) because things are changing over time. 3) Fill in the things you know and solve for the unknown.

Exploring Problem # 1 A 6 meter ladder is leaning against a wall. If the bottom of the ladder is pushed/pulled out at a constant rate of ½ meters per second, what else will be changing? GSP Demo Discuss some surprising observations:

NOTES—Problem #1 A 6 meter ladder is leaning against a wall. If the bottom of the ladder is pushed/pulled out at a constant rate of ½ meters per second, what else will be changing? How fast is the ladder sliding down the wall when the top of the ladder is 5 meters up the wall?

Exploring Problem #2 A winch (altitude 20 ft) reels in a rope at 2 ft/sec. How fast is the boat moving when the rope is 45 feet long? GSP Demo Discuss some surprising observations:

Problem #2 A winch (altitude 20 ft) reels in a rope at 2 ft/sec. How fast is the boat moving when the rope is 45 feet long?

Exploring Typical Example #2 (Similar Triangles Use Proportions)—Packet p. 4 A man 6 feet tall walks away from a lamp post 15 feet tall at a rate 0f 5 ft/sec. How fast is his shadow lengthening?

Is it the same as the rate that the shadow is lengthening? Discuss with your partner. GSP Demo A man 6 feet tall walks away from a lamp post 15 feet tall at a rate 0f 5 ft/sec. How fast is the shadow’s tip moving?

Typical Example #3 Cone Problems (Use a similar triangles sub-relationship) Water is flowing into a cone (height = 16 cm and radius = 4 cm) at a rate of 2 cubic cm per minute. How fast is the water level rising when the water level is 5 cm deep? What does 2 cubic cm per minute represent? ___________________ We will need the equation for the volume of cone…anyone remember what this would be? ____________

Exploring the Problem Water is flowing into a cone (height = 16 cm and radius = 4 cm) at a rate of 2 cubic cm per minute. How fast is the water level rising when the water level is 5 cm deep? GSP Demo SOLUTION

Water is flowing into a cone (height = 16 cm and radius = 4 cm) at a rate of 2 cubic cm per minute. How fast is the water level rising when the water level is 5 cm deep? Continued

Start on Packet p. 7