One model of Oil production in a certain country is given by the following equat

Question

One model of Oil production in a certain country is given by the following equation, where P(t) is the number of barrels of oil in billions, produced in a year, t years after 1910. P(t) = 0.0000000216t^4 - 0.0000167t^3 + 0.00156t^2 + 0 002t + 0 22, 0 lessthanorequalto t lessthanorequalto 105 According to this model, what is the absolute maximum amount of oil produced in the country and in what year did that production occur? According to this model, at what rate was the country's oil production declining in 2005 and in 2008? According to this model, the absolute maximum amount of oil produced in the country was billion barrels in the year. (Type an integer or decimal rounded to three decimal places as needed. Round down to the nearest year as needed.) In 2005, the country's oil production declined at a rate of billon barrels/yr. (Type an integer or decimal rounded to four decimal places as needed.) In 2008, the country's oil production was declining at a rate of billion barrels/yr. (Type an integer or decimal rounded to four decimal places as needed.)

Explanation / Answer

a)given P(t)=0.0000000216t4-0.0000167t3+0.00156t2+0.002t+0.22 , 0<=t<=105

differentiate with respect to t

P'(t)=0.0000000216*4t3-0.0000167*3t2+0.00156*2t+0.002+0

P'(t)=0.0000000864t3-0.0000501t2+0.00312t+0.002

for critical points P(t)=0

0.0000000864t3-0.0000501t2+0.00312t+0.002=0

t=71.6973

P(0)=0.22

P(71.6973) =2.7984

P(150)=-9.8075

t =71.6973

=>year =1910+71.6973=1981.6973

absolute maximum amount of oil produced in the country was 2.798 billion barrels in the year 1982

b) in the year 2005, t =2005-1910 =95

P'(95)=0.0000000864*953-0.0000501*952+0.00312*95+0.002

P'(95)=-0.0796753

oil production declined at rate of 0.0797 billion barrels per year

c) in the year 2008, t =2008-1910 =98

P'(98)=0.0000000864*983-0.0000501*982+0.00312*98+0.002

P'(98)=-0.09208

oil production declined at rate of 0.0921 billion barrels per year

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