RELATED RATES Section 2.6.

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Presentation transcript:

RELATED RATES Section 2.6

When you are done with your homework, you should be able to… Find a related rate Use related rates to solve real-life problems

Find the derivative of with respect to y None of the above

Find the volume of a cone with a radius of 24 inches and a height of 10 inches. Round to the nearest hundredth. 6031.86 0.03

FINDING RELATED RATES We use the chain rule to implicitly find the rates of change of two or more related variables that are changing with respect to time.

Some common formulas used in this section Volume of a… Sphere: Right Circular Cylinder: Right Circular Cone: Rectangular Pyramid: Pythagorean Theorem:

GUIDELINES FOR SOLVING RELATED-RATE PROBLEMS Identify all given quantities and quantities to be determined. Make a sketch and label the quantities. Write an equation involving the variables whose rates of change either are given or are to be determined. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t. After completing step 3, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change.

Find the rate of change of the distance between the origin and a moving point on the graph of y = sin x if dx/dt = 2 cm/sec.

Find the rate of change of the volume of a cone if dr/dt is 2 inches per minute and h = 3r when r = 6 inches. Round to the nearest hundredth. How is this problem different? 678.68 0.03

Vertical Motion. A ball is dropped from a height of 100 feet Vertical Motion. A ball is dropped from a height of 100 feet. One second later, another ball is dropped from a height of 75 feet. Which ball hits the ground first? Do we need to use calculus to solve this problem?

Conclusion: The first ball will hit the ground first, and the second ball will hit the ground second later.

Angle of Elevation. A fish is reeled in at a rate of 1 foot per second from a point 10 feet above the water. At what rate is the angle between the line and the water changing when there is a total of 25 feet of line out?

Consider the following situation: A container, in the shape of an inverted right circular cone, has a radius of 5 inches at the top and a height of 7 inches. At the instant when the water in the container is 6 inches deep, the surface level is falling at the rate of -1.3 in/s. Find the rate at which the water is being drained.