Question

Oil leaks out of a tanker at a rate of r=f(t) liters per minute, where t is in minutes. If f(t)=Ae^(−kt), write a definite integral expressing the total quantity of oil which leaks out of the tanker in the first hour.

Answer 1

The definite integral expressing the total quantity of oil, 'V', which leaks out of the tanker in the first hour is given as follows;

`V = \int\limits^(60)_0 {A \cdot e^((-k \cdot t))} \, dt`

Explanation:

From the question, we have;

The rate at which oil leaks out of the tanker, r = f(t)

The unit of the oil leak = Liters per minute

The unit of t = Minutes

`If \ f(t) = A \cdot e^((-k \cdot t ))`

Therefore, we have;

The definite integral expressing the total quantity, 'V', of oil which leaks out of the tanker in the first hour is given as follows;

`V = \int\limits^(60)_0 {A \cdot e^((-k \cdot t))} \, dt`

Therefore, we have;

`\int\limits^(60)_0 {A \cdot e^((-k \cdot t))} \, dt = (A \cdot e ^(60 \cdot k) - A)/(k \cdot e ^(60 \cdot k) )`