Based on a preliminary report by a geological survey team, it is estimated that

Question

Based on 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) = 700t^2/t^3 + 32 + 5 (0 lessthanorequalto t lessthanorequalto 20) thousand barrels/year, t years after production begins. Find the amount of oil that the field can be expected to yield during the first 8 years of production, assuming that the projection holds true. (Round your answer to the nearest thousand barrels.) thousand barrels The increase in carbon dioxide (CO_2) in the atmosphere is a major cause of global warming. Using data obtained by Charles David Keeling, professor at Scripps Institution of Oceanography, the average amount of CO_2 in the atmosphere from 1958 through 2010 is approximated by A (t) = 0.012313t^2 + 0.7545t + 313.9 (1 lessthanorequalto t lessthanorequalto 53) where A (t) is measured in parts per million volume (ppmv) and t in years, with t = 1 corresponding to 1958. t Find the average amount of CO_2 in the atmosphere from 1958 through 2010. ppmv

Explanation / Answer

Solution 13.

To find the amount we simply need to integrate the rate with respect to time
R(t) dt

= 1000 [ 700t² / (t³ + 32) ] + 5 dt from t = 0 to t = 8

= 1000 [ 700t² / (t³ + 32) ] dt + 5 dt  from t = 0 to t = 8

let u = t³ + 32 such that [ 700t² / u ] + 5 dt

du = 3t² dt ==> dt = du /(3t²)

= 1000 [ 700t² / (t³ + 32) ] dt + 1000 5 dt  from t = 0 to t = 8

= 1000 [ 700t² / u ] * du/(3t²) ]+ 10005 dt from t = 0 to t = 8

= 700000/3 1/u du +1000 5dt  from t = 0 to t = 8

= (700000/3)ln(u) + 5000t  from t = 0 to t = 8

= (700000/3)ln(t³ + 32) + 5000t  from t = 0 to t = 8

= 701083.11 barrels

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