Emily

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Emily

  1. 1. h r Page 223 - Problem 7 “ Let V be the volume of a cylinder having height h and radius r, and assume that h and r vary with time.” By Emily Selinger
  2. 2. Part A “ How are dV/dt, dh/dt, and dr/dt related?” The first derivative will include all of these terms, and therefore relate them, so we must find dV/dt -->
  3. 3. V = h·πr 2 dV/dt = h·2πr(dr/dt) + πr (dh/dt) 2 Use product rule to find first derivative:
  4. 4. Factor out π: dV/dt = π (h·2r ( dr/dt ) + r ( dh/dt ) ) 2 dV/dt = π (2rh ( dr/dt ) + r ( dh/dt ) ) 2
  5. 5. Part B h=6 r=10 “ At a certain instant, the height is 6 inches and increasing at 1 in/s, while the radius is 10 inches and decreasing at 1 in/s. How fast is the volume changing at that instant? Is the volume increasing or decreasing at that instant?”
  6. 6. We are trying to find out how fast the volume is changing, which means we have to find dV/dt: V = h·πr 2 dV/dt = h·2πr(dr/dt) + πr (dh/dt) 2 And we now know that:
  7. 7. We also know that: h = 6 in r = 10 in dh/dt = 1 in/s dr/dt = -1 in/s (it is decreasing, therefore it’s negative) So we can sub these in: dV/dt = 6·2π(10)(-1) + π(10) (1) 2
  8. 8. dV/dt = -120π + 100π dV/dt = -20π *dV/dt is DECREASING. We know this because the answer comes out to be negative

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