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Appendix C C-25
either the quadratic formula or a graphing utility.) The root represents the
“life expectancy” of oil under this scenario. Round the answer to the nearest
ten years and add to 1990 to get a depletion date for oil.
A consequence of the quadratic yearly consumption model: Suppose
that the consumption y (in billions of barrels) in year x is given by a quadratic
function, y ax 2 bx c. Then, using a result from Chapter 14
(Exercise 5 in Section 14.1), it can be shown that the total amount A of oil
consumed over the years x 0 through x T is given by the cubic model
A a 3 T 3 a b T 2 a
2
a 3b 6c
b T c
6
On the left side of this equation, replace A with 2886; on the right side, replace
a, b, and c with the values that you obtained for the quadratic model
in part (a). Now use a graphing utility to find the positive root of the resulting
cubic equation. The root gives you the “life expectancy” of oil
under this scenario. Round the answer to the nearest ten years and add to
1990 to get a depletion date for oil.
Constant consumption model: If you’ve done the preceding calculations
correctly, you will have found that the depletion dates under both
models occur within the present century. Now try the following “what if.”
Go back to the linear consumption model and redo the calculations
using for m the value 0 rather than the value from part (a). In terms of
consumption, what’s the interpretation of m 0? What depletion date do
you obtain?
(d) Write a paragraph or two summarizing the results from part (c). (Don’t
forget to include what the assumptions are in each case.)