Suppose that r(t) = r0 * e^(-kt) with k > 0, is the rate at which a nation extra

Question

Suppose that r(t) = r0 * e^(-kt) with k > 0, is the rate at which a nation extracts oil where r0=10^7 barrels/year is the current rate of extraction. Suppose also that the estimate of the total oil reserve is 2*10^9 barrels. a. Find Q(t), the total amount of oil extracted by the nation after t years. b. Evaluate lim(t->infinity)Q(t) and explain the meaning of this limit. c. Find the minimum decay constant k for which the total oil reserves will last forever. d. Suppose r0=2*10^7 barrels/yr and the decay constant k is the minimum value found in part c. How long will the total oil reserves last?

Explanation / Answer

r(t) = r0 * e^(-kt)
r0=10^7 barrels/year
total oil reserve =2*10^9 barrels.
a)

Q(t)=t=tt=0 r(t)

=ro (e^(-kt))

=ro( (1-e^(-k)*(t+1))/(1-e^(-k))

b)lim t-> Q(t) =ro/(1-e^(-k))

The total amount of oil is extracted after a very long time

c) ro/(1-e^(-k)) =2*10^9

=>1-e^(-k)=.5*10^-2

=>e^-k=.995

=>k=5.012*10^-3 /year

d)r0=2*10^7 barrels/yr

k=5.012*10^-3

2*10^9*(1-e^-K)=2*10^7 (1-e^(-k(t+1))

=>1-e^(-k(t+1))=.5

=>e^(k(t+1))=2

=>k(t+1)=.693

=>t=137.08 years

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