Question

An oil tanker breaks apart and starts leaking. As time goes on, the rate at which the oil is leaking out will diminish. Suppose that "t" hours after the tanker breaks apart, the oil is leaking out at a rate of R(t)=(0.7)/(1+t^2) million gallons per minute. Then ___ million gallons of oil will leak out in the first 3 hours after the shipwreck.

Answer 1

Answer:

62.77 million gallons

Step-by-step explanation:

Given,

The function that shows rate of leaking oil per minute in after t hours,

$R(t) = \frac{0.7}{1+t^2}$

∵ 1 hour = 60 minutes ⇒ 3 hours = 180 minutes,

Hence, the quantity of oil leaked out in the first 3 hours,

$V(t) = \int_{0}^{180} R(t) dt$

$=\int_{0}^{180} \frac{0.7}{1+t^2} dt$

$=0.7 \int_{0}^{180} \frac{dt}{1+t^2}$

$=0.7[tan^{-1}(t)]_{0}^{180}$

$=0.7(tan^{-1}(180) - tan^{-1} (0))$

$=0.7(89.68)$

= 62.77 million gallons