(a) To find the total number of barrels leaked on the first day, we need to integrate the function from t=0 to t=24 (24 hours in a day):
∫[0,24] (66ln(t+1))/(t+1) dt = 66∫[0,24] ln(t+1) dt
= 66[tln(t+1) - t] | from 0 to 24
= 66[(24ln(24+1) - 24) - 0]
= 66[(24ln(25) - 24)]
≈ 527.39 barrels
Therefore, the ship will leak approximately 527.39 barrels of oil on the first day.
(b) To find the total number of barrels leaked on the second day, we need to integrate the function from t=24 to t=48:
∫[24,48] (66ln(t+1))/(t+1) dt = 66∫[24,48] ln(t+1) dt
= 66[tln(t+1) - t] | from 24 to 48
= 66[(48ln(48+1) - 48) - (24ln(24+1) - 24)]
≈ 879.61 barrels
Therefore, the ship will leak approximately 879.61 barrels of oil on the second day.
(c) As t approaches infinity, the function L(t) approximates to 66. This means that over the long run, the amount of oil leaked per day will approach 66 barrels.
An oil tanker is leaking oil at a rate given in barrels per hour by the function shown below where tea is the time and hours after the tanker hits a hidden rock (when t=0). Complete parts (a) through (c). L(t)=(66ln(t+1))/(t+1)
(a) Find the total number of barrels that the ship will leak on the first day.
(b) Find the total number of barrels that the ship will leak on the second day.
(c) what is happening over the long run to the amount of oil leaked per day?
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