# 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...

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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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.*