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)=St=tt=0 r(t) =ro S(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

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.