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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.
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.
Final answer: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.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....