Questions: Oil is leaking out of a ruptured tanker at the rate of r(t)=20 e^(-0.04 t) thousand liters per minute. A. At what rate, in thousands of liters per minute, is the oil leaking out at t=0 ? rate =20 thousand liters / min at t=60 ? rate = thousand liters/min B. How many thousands of liters leak out during the first hour? Number of liters = thousand liters

Solution Steps

To find the leak rate at \( t = 0 \), we evaluate the rate function \( r(t) \): \[ r(0) = 20 e^{-0.04 \cdot 0} = 20 \text{ thousand liters/min} \]

Next, we calculate the leak rate at \( t = 60 \): \[ r(60) = 20 e^{-0.04 \cdot 60} \approx 1.8144 \text{ thousand liters/min} \]

To find the total amount of oil leaked during the first hour, we integrate the rate function from \( t = 0 \) to \( t = 60 \): \[ \text{Total leaked} = \int_0^{60} r(t) \, dt = \int_0^{60} 20 e^{-0.04 t} \, dt \approx 454.6410 \text{ thousand liters} \]

Final Answer