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AP Calculus Name Review Related Rates 1. Suppose the radius of a spherical balloon is shrinking at ½ cm per minute. How fast is the volume decreasing when the radius is 4 cm? 2. Suppose a snowball remains spherical while it melts, with the radius shrinking at 1 inch per hour. How fast is the volume of the snowball decreasing when the radius is 2 inches? 3. Suppose the volume of the snowball in problem 2 shrinks at the rate of 2/V (cubic inches per hour), so that dV/dt = -2/V. How fast is the radius changing when the radius is ½ inch? 4. A spherical balloon is inflated at the rate of 3 cubic centimeters per minute. How fast is the radius of the balloon increasing when the radius is 6 cm? 5. Suppose a spherical balloon grows in such a way that after t seconds, V = 4 t cm3. How fast is the radius changing after 64 seconds? 6. A spherical balloon is losing air at the rate of 2 cm3/min. How fast is the radius of the balloon shrinking when the radius is 8 cm? 7. Water leaking onto a floor creates a circular pool with an area that increases at the rate of 3 cm2/min. How fast is the radius of the pool increasing when the radius is 10 cm? 8. A point moves around the circle x2 + y2 = 9. When the point is at (- 3 , 6 ), its x coordinate is increasing at the rate of 20 units per second. How fast is its y coordinate changing at that instant? 9. A ladder 15 feet long leans against a vertical wall. Suppose that when the bottom of the ladder is x feet from the wall, the bottom is being pushed toward the wall at the rate of ½ x ft/sec. How fast is the top of the ladder rising at the moment the bottom is 5 feet from the wall? 10. A board is 5 feet long slides down a wall. At the instant the bottom end is 4 feet from the wall, the other end is moving down the wall at the rate of 2 ft/sec. At that moment: a) How fast is the bottom end sliding along the ground? b) How fast is the area of the region between the board, ground, and wall changing? 11. A water trough is 12 feet long, and its cross section is an equilateral triangle with sides 2 feet long. Water is pumped into the trough at a rate of 3 ft3/min. How fast is the water level rising when the depth of the water is ½ foot? 12. A beacon on a lighthouse 1 mile from shore revolves at the rate of 10 rad/min. Assuming that the shoreline is straight, calculate the speed at which the spotlight is sweeping across the shoreline as it lights up the sand 2 miles from the lighthouse. (Hint: dx/dt is the speed of the image of the spotlight moving across the shoreline , and d /dt = 10 ) 13. Boyle’s Law states that if the temperature of a gas remains constant, then the pressure p and the volume v of the gas satisfy the equation pv=c where c is a constant. If the volume is decreasing at the rate of 10 cm3/sec, how fast is the pressure increasing when the pressure is 100 lbs/cm 2 and the volume is 20 cm3? Answers: 1) DV/dt = -32 cm3/min 2) DV/dt = -16in3/hr 3) Dr/dt = -12/ in/h 4) Dr/dt = 1/48 cm/min 1 cm 5) Dr/dt = s 1 2 64( π 3 )(3 3 ) 1 cm 6) Dr/dt = 128π min 3 cm 7) Dr/dt = 20π min 8) Dy/dt = 10 2 u sec 5 2 ft 9) Dy/dt = sec 8 3 ft 2 10) a) b) dA 7 in sec 2 sec dt 4 27 in 11) dh/dt = 8π sec 3 ft 12) dh/dt = 4 min 13) dx/dt = 40 mi/min 14) dp/dt = 50 lbs/cm2/sec

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posted: | 10/20/2011 |

language: | English |

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