# Oil spilled from a ruptured oil tanker spreads out in a circular fashion. If the diameter of the spill increases at a constant rate of 1/4 mile/ hr. how fast is the area of the spill increasing...

Oil spilled from a ruptured oil tanker spreads out in a circular fashion. If the diameter of the spill increases at a constant rate of 1/4 mile/ hr. how fast is the area of the spill increasing when the area is 100 square miles ?

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You need to notice that the problem provides the rate the diameter of the spill increases such that:

`(dD)/(dt) = 1/4` miles/hour

You need to write the area of the circle in terms of diameter such that:

`A = pi*D^2/4`

You need to differentiate the area with respect to t such that:

`(dA)/(dt) = 2piD/4*(dD)/(dt) `

You need to evaluate the diameter if `A = 100` square miles such that:

`A = pi*D^2/4 =gt 100 = pi*D^2/4 =gt 400 = pi*D^2`

`200/sqrt pi= D`

You need to substitute `200/sqrt pi` for D and `1/4` for `(dD)/(dt)` in `(dA)/(dt) = 2piD/4*(dD)/(dt)` such that:

`(dA)/(dt) = 2pi 200/(4sqrt pi)*(1/4)`

`(dA)/(dt) = 25 sqrt pi` square miles per hour.

**Hence, evaluating how fast the area of the spill increasing under given conditions yields `(dA)/(dt) = 25 sqrt pi` square miles per hour.**

If 1/4 spills in 1 hour, you have to divide 100 by 1/4! which would give 400 hours