5.3.3 Calculus AB Independent Study Finding Rates of Changes Study Sheet Note: The important things to get here are: how to recognize a rate when you see one the importance of units in the process of recognizing and stating rates of change that decreasing quantities have negative rates 1. A tank of water is in the shape of a cone, point down, with radius 4 feet and height 10 feet. The water is draining out of the tank at a rate of .2 cubic feet per minute. How fast is the level of the water falling when the water is 8 feet deep? A. What rate is given in the problem? Your answer must include a description in words (rate of change of volumewith respect to time), the number and units, mathematical symbols for the rate, and definitions of the letters you've chosen for your variables. (For example, your answer to a different problem might be "velocity or change in distance with respect to time, 50 mph, ds, where s is the distance traveled and : is time.") B. What rate is asked for? Your answer must include a description in words (rate of change of height with respect to time ), the units, mathematical symbols for the rate, and definitions of the letters you've chosen for your variables. (For example, your answer to a different problem might be "rate of change of distance traveled with respect to volume of gasoline or miles per gallon, , where s is the distance traveled in miles and I is the volume of the gasoline, in gallons.") 2. How fast does the radius of a spherical soap bubble change when you blow air into it at the rate of 15 cubic centimeters per second? A. What rate is given in the problem? Your answer must include a description in words (rate of change of volume, with respect to time ), the number and units, mathematical symbols for the rate, and definitions of the letters you've chosen for your variables. B. What rate is asked for? Your answer must include a description in words (rate of change of radius with respect to time ), the units, mathematical symbols for the rate, and definitions of the letters you've chosen for your variables. 3. A baseball diamond is a square 90 feet on a side. A player runs from first base to second base at a rate of 15 feet per second. At what rate is the player's distance from third base decreasing when the player is halfway between first and second base? Note: a distance must be between two things. A. What rate is given in the problem? Your answer must include a description in words (rate of change of with respect to time), the number and units, mathematical symbols for the rate, and definitions of the letters you've chosen for your variables. the distance between the player and second base B. What rate is asked for? Your answer must include a description in words (rate of change of with respect to time ), the units, mathematical symbols for the rate, and definitions of the letters you've chosen for your variables. the distance between the player and third base