Related RatesNovember 5th, 2012

1. Oil spilled from a ruptured tanker spreads in a circle whose area increasesat a constant rate of 6 mi2/h. How fast is the radius of the spill increasingwhen the area is 9 mi2?

2. A softball diamond is a square whose sides are 60 ft long. Suppose that aplayer running from first to second base has a speed of 25 ft/s at the instantwhen she is 10 ft from second base. At what rate is the players distance fromhome plate changing at that instant?

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3. On a dark night, a 6 ft tall man walks away from a lamp post 24 ft high ata rate of 4 ft/sec. How fast is the end of his shadow moving? How fast is hisshadow lengthening?

4. A conical water tank has a height of 20 ft and a radius of 10 ft at the top.If water is being poured into the tank at a rate of 10 ft3/min, how quicklyis the depth of the water increasing when the water is 8 ft deep? when thewater is 12?

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5. A beacon that makes one revolution every 10 seconds is on a ship anchored4 km from a straight shoreline. How fast is the beam moving along theshoreline when it makes an angle of 45◦ with the shoreline.

6. A camera is mounted 4000 ft from the base of a rocket launching pad. If therocket is rising vertically at 1000 ft/sec when it is 3000 ft above the launchingpad, how fast must the camera elevation angle change at that instant to keepthe rocket in sight? How fast is the camera-rocket distance changing at thatsame moment?

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