Ch 4.6 Related Rates Graphical, Numerical, Algebraic by Finney Demana, Waits, Kennedy

1. Ch 4.6 Related RatesGraphical, Numerical, Algebraic by Finney Demana, Waits, Kennedy

4. Differentiation With Respect to Different Variables The volume of a cone can be expressed as: a) Find dV/dt if V, r, and h are all functions of time. b) Find dV/dt if r is constant so that V and h are functions of time. c) Find dV/dt if V is constant so that only r and h are functions of time.

5. Differentiation With Respect to Different Variables The volume of a cone can be expressed as: a) Find dV/dt if V, r, and h are all functions of time. b) Find dV/dt if r is constant so that V and h are functions of time. c) Find dV/dt if V is constant so that r and h are functions of time.

6. Related Rate Example A hot-air balloon rising straight up from a level field is tracked by a range finder 500 feet from the lift-off point. At the moment the range finder’s elevation angle is π/4, the angle is increasing at the rate of .14 radians/min. How fast is the balloon rising at that moment?

7. h R.F. 500 ft Related Rate Example A hot-air balloon rising straight up from a level field is tracked by a range finder 500 feet from the lift-off point. At the moment the range finder’s elevation angle is π/4, the angle is increasing at the rate of .14 radians/min. How fast is the balloon rising at that moment?

8. Steps to Solving a Related Rate Problem • Identify the variable whose rate of change we want and the variables whose rate of change we have. • Draw a picture and label with values and variables. • Write an equation relating the variable wanted with known variables. • Differentiate implicitly with respect to time. • Substitute known values • Interpret the solution and reread the problem to make sure you’ve answered all the questions.

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12. Related Rate Example Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?

13. 5 ft r 10 ft h Related Rate Example Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?

14. Related Rate Example • A 13 ft ladder is leaning against a wall. Suppose that the base of the ladder slides away from the wall at the constant rate of 3 ft/sec. • Show how the motion of the two ends of the ladder can be represented by parametric equations. • What values of t make sense in this problem. • Graph using a graphing calculator in simultaneous mode. State an appropriate viewing window. (Hint: hide the coordinate axis). • Use analytic methods to find the rates at which the top of the ladder is moving down the wall at t = .5, 1, 1.5 and 2 sec. How fast is the top of the ladder moving as it hits the ground?

15. Related Rate Example • A 13 ft ladder is leaning against a wall. Suppose that the base of the ladder slides away from the wall at the constant rate of 3 ft/sec. • Show how the motion of the two ends of the ladder can be represented by parametric equations. • What values of t make sense in this problem. • { 0 < t < 13/3 }

16. Related Rate Example • Graph using a graphing calculator in simultaneous mode. State an appropriate viewing window. (Hint: hide the coordinate axis). t: [0,4] x: [0,13] y: [0, 13] • 4. Use analytic methods to find the rates at which the top of the ladder is moving down the wall at t = .5, 1, 1.5 and 2 sec. How fast is the top of the ladder moving as it hits the ground?