Question

An oil tanker breaks apart and starts leaking. Suppose that right after the shipwreck, the oil is leaking out at a rate of 0.1 million gallons per minute, and that the rate decays exponentially with a half-life of 110 minutes. Then, t minutes after the shipwreck the oil is leaking at a rate of R(t) = __________ million gallons per minute, and __________ million gallons of oil will leak out in the first 330 minutes after the shipwreck.

Answer 1

The rate of oil leakage after the shipwreck can be calculated using an exponential decay function. The rate of oil leakage, R(t), t minutes after the shipwreck is given by R(t) = (Initial Rate) * (0.5)^(t / Half-life). The total amount of oil leaked in the first 330 minutes can be calculated by integrating the rate function, R(t), from 0 to 330.

Step-by-step explanation:

The rate of oil leaking after the shipwreck can be modeled by an exponential decay function. Given that the half-life of the oil leak is 110 minutes, the rate of oil leakage, R(t), at any given time, t, can be calculated using the formula:

R(t) = (Initial Rate) * (0.5)^(t / Half-life)

Plugging in the values, the rate of oil leakage, R(t), t minutes after the shipwreck is 0.1 * (0.5)^(t / 110) million gallons per minute. To calculate the total amount of oil leaked in the first 330 minutes, we need to integrate the rate function, R(t), from 0 to 330:

Total amount of oil = ∫[0, 330] (R(t) dt)

Simplifying and integrating the function, we find that approximately 0.191 million gallons of oil will leak out in the first 330 minutes after the shipwreck.