Question

A 100-gallon barrel, initially half-full of oil, develops a leak at the bottom. Let A(t) be the amount of oil in the barrel at time t. 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. The mathematical model is

Answer 1

`\frac{\mathrm{d} A}{\mathrm{d} t}=ktA\,,\,A(0)=50`

Explanation:

A(t) denotes the amount of oil in the barrel at time t.

As 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,

`\frac{\mathrm{d} A}{\mathrm{d} t}=ktA`

Here, k is a constant term.

As a 100-gallon barrel initially contains half-full of oil,

`A(0)=50`