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differentiating relative rates: a stone dropped into a still pond sends out a circular...

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user7230927 | (Level 1) Salutatorian

Posted July 11, 2013 at 3:41 AM via web

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differentiating relative rates:

a stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 5 ft/sec.  how rapidly is the area enclosed by the ripple increasing at the end of 16 sec?

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jeew-m | College Teacher | (Level 1) Educator Emeritus

Posted July 11, 2013 at 3:55 AM (Answer #1)

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Let us say the radius of the circular ripple at any point is rft. It is given that the rate of increasing of r is 5ft/s.

`(dr)/(dt) = 5`

 

If the area of the circular ripple is A then;

`A = pir^2`

By differentiation;

`(dA)/(dt) = pixx2rxx(dr)/(dt)`

 

When the time is 16 seconds;

`r = txx(dr)/(dt)`

`r = 16xx5 = 80ft`

 

`((dA)/(dt))_(t = 16) = pixx2rxx(dr)/(dt)`

`((dA)/(dt))_(t = 16) = pixx2xx80xx5`

`((dA)/(dt))_(t = 16) = 800pi`

 

So the area is increasing at a rate of `800pift/s` when the time is 16 seconds.

 

Assumptions

  • At t = 0 the radius of the circular ripple is zero or negligible
  • The radius of the pond is larger than 80ft.

 

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