4.6 RELATED RATES. STRATEGIES FOR SOLVING RELATED RATES PROBLEMS 1.READ AND UNDERSTAND THE PROBLEM. 2.DRAW AND LABEL A PICTURE. DISTINGUISH BETWEEN CONSTANT.

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Presentation transcript:

4.6 RELATED RATES

STRATEGIES FOR SOLVING RELATED RATES PROBLEMS 1.READ AND UNDERSTAND THE PROBLEM. 2.DRAW AND LABEL A PICTURE. DISTINGUISH BETWEEN CONSTANT QUANTITIES AND VARIABLES THAT CHANGE OVER TIME. 3.WRITE AN EQUATION RELATING THE VARIABLE WHOSE CHANGE YOU WANT WITH THE VARIABLES WHOSE CHANGE YOU KNOW. 4.DIFFERENTIATE BOTH SIDES OF THE EQUATION. 5.SUBSTITUTE VALUES INTO EQUATION. 6.INTERPRET THE SOLUTION.

EXAMPLE 1:FINDING RELATED RATE EQUATIONS ASSUME THAT THE RADIUS R OF A SPHERE IS A DIFFERENTIABLE FUNCTION OF T AND LET V BE THE VOLUME OF THE SPHERE. FIND AN EQUATION THAT RELATES DV/DT AND DR/DT.

ASSUME THAT THE RADIUS R AND HEIGHT H OF A CONE ARE DIFFERENTIABLE FUNCTIONS OF T AND LET V BE THE VOLUME OF THE CONE. FIND AN EQUATION THAT RELATES DV/DT, DR/DT, AND DH/DT.

EXAMPLE 2: A RISING BALLOON

EXAMPLE 3: A HIGHWAY CHASE A POLICE CRUISER, APPROACHING A RIGHT-ANGLED INTERSECTIONS FROM THE NORTH, IS CHASING A SPEEDING CAR THAT HAS TURNED THE CORNER AND IS NOW MOVING STRAIGHT EAST. WHEN THE CRUISER IS 0.6 MILES NORTH OF THE INTERSECTION AND THE CAR IS 0.8 MILES TO THE EAST, THE POLICE DETERMINE WITH RADAR THAT THE DISTANCE BETWEEN THEM AND THE CAR IS INCREASING AT 20 MPH. IF THE CRUISER IS MOVING AT 60 MPH AT THE INSTANT OF MEASUREMENT, WHAT IS THE SPEED OF THE CAR?

EXAMPLE 4: FILLING A CONICAL TANK WATER RUNS INTO A CONICAL TANK AT THE RATE OF 9 FT 3 /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?