An oil pipeline suddenly breaks at time t=0 and oil leaks out at a rate of r(t)=52e−0.01t thousand liters per minute.

How many thousand liters of oil leaks out in the first 15 minutes? Round to the nearest thousand liters.

Question

An oil pipeline suddenly breaks at time t=0 and oil leaks out at a rate of r(t)=52e−0.01t thousand liters per minute.

How many thousand liters of oil leaks out in the first 15 minutes? Round to the nearest thousand liters.

BlackTom AI · Accepted Answer

We are given a leak rate r(t) = 52 e^(−0.01 t) thousand liters per minute, and we want the total oil leaked in the first 15 minutes. This is a accumulation problem, so we integrate the rate over the time interval from t = 0 to t = 15.

Step 1: Set up the integral. The total volume leaked is V = ∫ from 0 to 15 of r(t) dt = ∫_0^{15} 52 e^{−0.01 t} dt, in thousand liters.

Step 2: Compute the integral. The antiderivative of e^{−0.01 t} with respect to t is (−1/0.01) e^{−0.01 t} = −100 e^{−0.01 t}.
Thus ∫_0^{15} 52 e^{−0.01 t} dt = 52 [−100 e^{−0.01 t}]_0^{15} = 52 [−100 e^{−0.15} − (−100 e^{0})] = 5200 [1 − e^{−0.15}].

Step 3: Evaluate e^{−0.15}. Numerically, e^{−0.15} ≈ 0.8607, so 1 − e^{−0.15} ≈ 0.1393.
Therefore V ≈ 5200 × 0.1393 ≈ 724.3 thousand liters.

Step 4: Apply the requested rounding. Rounding to the nearest thousand liters gives about 724 thousand liters.

Note on units and interpretation: since the rate is in thousand liters per minute and the time is in minutes, the integral yields thousand liters, which is consistent with the requested rounding unit (thousand liters).

If you compare to approximate methods, you could also use the fact that the integral of an exponential decays toward the total potential amount 5200 thousand liters, with the time constant 100 minutes, and the interval 15 minutes is a small fraction of that, but the direct calculation above yields a precise rounded result.

An oil pipeline suddenly breaks at time t=0 and oil leaks out at a rate of r(t)=52e−0.01t thousand liters per minute. How many thousand liters of oil leaks out in the first 15 minutes? Round to the nearest thousand liters.

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