Related Rates STEP 1 STEP 2:

Transcription

1 Related Rates You can use derivative analysis to determine how two related quantities also have rates of change which are related together. I ll lead off with this example. 3 Ex) A spherical ball is being filled with air at a rate of 150in /min. a) State the equation which relates the volume of a sphere to its radius. b) Differentiate both sides with respect to time. c) At the instant the radius is 6-in long, at what rate is the radius expanding? SOLUTION: This solution will offer a general outline strategy for solving problems relating rates of change. STEP 1: READ AND RE-READ THE PROBLEM CAREFULLY and using either geometry or possibly a diagram of the situation in the problem, determine a formula which relates the quantities together. STEP 2: Identify which of the values used in the formula from step 1 which have changing rates and differentiate them with respect to time t.

2 STEP 3: RE-re-read the problem to see (1) what you re actually asked to find and (2) what values are known and what values are unknown to substitute into your rate relation formula from step 2. The radius is growing at a rate of at this instant. STEP 4: Make sure your answer makes sense. Compare the rates of change at the instant the radius is r = 1 inch r = 10 inches

3 Ex) Water is being pumped into a tank shaped like a right circular cylinder. The tank has a radius of 2 meters and a height of 5 meters. The water level of a cylindrical tank is constantly rising at a rate of 0.3 meters per minute. How many cubic meters of water are being pumped into the tank per minute? STEP 1: Determine a relevant formula

4 Ex) A 20-ft. ladder is propped against a wall. At time t = 0, the base of the ladder is located 5- ft away from the base of the wall. Then the ladder s base begins sliding away from the wall at a rate of 2-ft/sec. How fast is the top of the ladder sliding down the wall after 2 seconds? SOLUTION: STEP 1: Determine a relevant formula This one might require a picture.

5 Ex) Two cars leave an intersection at the same time one heading south and the other heading east, both along very straight sections of highway (they re in Kansas). The southbound car travels at a speed of 50 mph and the east bound car travels at a speed of 70 mph. After 12 minutes of travel time, how fast is the distance between the two cars increasing (i.e. what is the speed in mph of the increasing distance between the two cars after 12 minutes)? STEP 1: Determine a relevant formula This one might require a picture

6 Ex) On a very clear day, you take your telescope out to watch the launching of a rocket. You have to position your telescope a distance of 10-km away from the launch site, but the ground between your telescope and the launch site is relatively level. After the rocket is launched, you begin to tilt your telescope to keep the rocket in your view. At the instant your telescope is tilted at an angle of π /3 radians, you also notice that you re tilting your telescope at a rate of 0.5 radians/minute. At this same instant, how fast is the rocket moving upward? STEP 1: Determine a relevant formula This one might require a picture

7 Ex) A tank shaped like an inverted right circular cone is being drained at a rate of 10 cubic feet per minute. The height of the tank is 3 times its radius. a) At the instant water level is 12 feet deep, how fast is the water level decreasing? b) How fast is the water level decreasing when the water level drains to 6 feet deep? STEP 1: Determine a relevant formula This one might require a picture