How do you intepret this problem?

Question

Can you help me with this calculus problem?
Oil Depletion, Suppose the amount of oil pumped from one of the canyon wells in Whittier, California, decreases at the continuous rate of 10% per year. When will the well's output fall to one-fifth of its present level?

Answer 1

oil = original * (0.9)^t

t is in years

What you want is to find t such that
(0.9)^t = 0.2

log (base 0.9) each side:

t = log (base 0.9) 0.2

Answer 2

Let x0 be the amount of oil pumped at year zero
Next year x1 = x0 - 0.1x0 = x0(0.9)
Next year x2 = x1 - 0.1x1 = x1(0.9) = 0.9(0.9)x0
.
.
.
xn = xn-1 - 0.1xn-1 = xn-1(0.9) = (0.9)^(n)x0
If xn = 0.2x0 = (0.9)^(n)x0, what is n, the number of years
The x0's cancel so
0.9^n = 0.2
ln[0.9^n] = ln[0.2]
n ln[0.9] = ln[0.2]
n = ln[0.2]/ln[0.9] where n is the number of years

Answer 3

Maybe soem geniuses know.

Answer 4

0.1 + 0.09 + 0.009 + 0.0009 + ...
= 0.1 + 0.09*1/(1 -- 0.1)
= 0.1 + 0.1
= 0.2
Never it will reach one - fifth.