On the basis of a preliminary report by a geological survey team, it is estimate

Question

On the basis of a preliminary report by a geological survey team, it is estimated that a newly discovered oil field can be expected to produce oil at the rate of R(t) = 600t^2/t^3 + 32 + 5 (0 lessthanorequalto t lessthanorequalto 20) thousand barrels/year, t years after production begins. a) Write an integral expression that will allow you to find the amount of oil that the field can be expected to yield during the first 5 years of production, assuming that the projection holds true. b) Use your calculator to evaluate the expression.

Explanation / Answer

The position of an arm is given by,

s = [(30) t2 - (20) t3] mm                                                             { eq.1 }

differentiating an above eq. w.r.t time, we have

v = [(60) t - (60) t2] mm/s    { eq.2 }

again, differentiating eq.2 w.r.t time & we get -

a = [(60) - (120) t] mm/s2

(a) When (dv / dt) = a = 0. Then, we have

0 = [(60) - (120) t] mm/s2

t = 1/2 sec

At this time, the maximum velocity will be given as :

v = [(60) t - (60) t2] mm/s

v = [(60) (0.5 s) - (60) (0.5 s)2] mm/s

v = [(30) - (15)] mm/s

v = 15 mm/s

(b) When velocity is maximum, then the position and acceleration will be given as :

s = [(30) t2 - (20) t3] mm     

s = [(30) (0.5)2 - (20) (0.5)3] mm    

s = [(7.5) - (2.5)] mm

s = 5 mm

And

a = [(60) - (120) t] mm/s2

a = [(60) - (120) (0.5)] mm/s2

a = 0 mm/s2

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