Recitation 8

Exercise: A cylinder is inscribed in a sphere of radius 10 cm. When the height of the cylinder is 10 cm and is increasing at 2 cm/min. How fast is the volume of the cylinder changing?

50\pi cm^3/min.

Exercise: A plane flying at an altitude of 1 mile is 2 miles distant from an observer, measured along the ground, and flying directly away from the observer at 400 mph. How fast is the angle of elevation changing?

-80 rad/hour

Exercise: A 10 foot ladder is leaning against a vertical wall. The bottom of the ladder is being pulled away from the wall at the rate 2 feet/sec.

  1. At what rate is the top of the ladder moving down the wall when the top is 6 feet from the ground?
  2. At what rate is the angle of the ladder with the ground changing then?
  3. At waht rate is the area of the triangle formed by the wall, the ground, and the ladder changing then?
  1. \frac{-8}{3} feet/sec
  2. \frac{-1}{3} rad/sec
  3. \frac{-14}{3} feet^2/sec

Exercise: A spherical dew drop is gaining volume through condensation. If the rate of change of the volume is proportional to its surface area, prove that the rate of change of its radius is constant.

As V=\frac{4}{3}\pi r^3, \frac{dV}{dt}=4\pi r^2\frac{dr}{dt}. Since \frac{dV}{dt}=4k\pi r^2, where k is a constant, \frac{dr}{dt}=k is a constant as well.

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