An oil storage tank ruptures at time t = 0 t=0 and oil leaks from the tank at a rate of r ( t ) = 75 e − 0.04 t(t)=75e-0.04to li
ters per minute. How much oil leaks out during the first hour
1 answer:
Answer:
Approximately 1705 liters of oil leaked out during the first hour.
Step-by-step explanation:
We are given the following in the question:
The rate of oil leak is given by:

This function gives the rate of leakage in liters per minute.
We have to find the amount of water leak during the first hour.
It can be calculated as:
![r(t) = 75 e^{-0.04t}\\\text{Integrating both sides}\\\\\displaystyle\int^{60}_0 = \int^{60}_0 75 e^{-0.04t}dt\\\\=\bigg[\dfrac{75 e^{-0.04t}}{-0.04}\bigg]^{60}_0\\\\=-1875\big[e^{-0.04(60)}-e^{0}\big]\\=-1875(0.0907-1)\\=1704.90\\\approx 1705](https://tex.z-dn.net/?f=r%28t%29%20%3D%2075%20e%5E%7B-0.04t%7D%5C%5C%5Ctext%7BIntegrating%20both%20sides%7D%5C%5C%5C%5C%5Cdisplaystyle%5Cint%5E%7B60%7D_0%20%3D%20%5Cint%5E%7B60%7D_0%2075%20e%5E%7B-0.04t%7Ddt%5C%5C%5C%5C%3D%5Cbigg%5B%5Cdfrac%7B75%20e%5E%7B-0.04t%7D%7D%7B-0.04%7D%5Cbigg%5D%5E%7B60%7D_0%5C%5C%5C%5C%3D-1875%5Cbig%5Be%5E%7B-0.04%2860%29%7D-e%5E%7B0%7D%5Cbig%5D%5C%5C%3D-1875%280.0907-1%29%5C%5C%3D1704.90%5C%5C%5Capprox%201705)
Thus, approximately 1705 liters of oil leaked out during the first hour.
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