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## Homework Statement

A 44 gallon barrel of oil develops a leak at the bottom. Let [itex]A(t)[/itex] be the amount of oil in the barrel at a given time [itex]t[/itex]. Suppose that the amount of oil is decreasing at a rate proportional to the product of the time elapsed and the amount of oil present in the barrel.

a. Give the mathematical model for A

b. Find the general solution of the differential equation

c. Find the particular solution for the initial condition

## Homework Equations

## The Attempt at a Solution

a.

[itex]\frac{\partial{A}}{\partial{t}} = -Akt[/itex]

b. [itex]\int{\frac{1}{A}\,dA} = \int{-kt\,dt}[/itex]

[itex]\ln{A}=\frac{kt^2}{2}+C[/itex]

[itex]A(t)=Ce^\frac{-kt^2}{2}[/itex]

c. [itex]\ln{A}=\frac{kt^2}{2}+C[/itex]

[itex]A(t)=Ce^\frac{-kt^2}{2}[/itex]

[itex]44=Ce^0[/itex]

[itex]A(t)=44e^\frac{-kt^2}{2}[/itex]

[itex]A(t)=44e^\frac{-kt^2}{2}[/itex]

Is this correct? I'm not sure if I came up with the right [itex]\frac{\partial{A}}{\partial{t}}[/itex].

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