Page 17 - Calculus for the Clueless

But e ln 9 = 9

Whew!!!!

In 50 = (t/4) In 9. So t = 4 In 50/ln 9 = 7.12 hours, by calculator.

The simpler way to get part 1A. If you notice the numbers, you will see that 9 comes from 90/10. Although the
time, 4 hours, originally is in the numerator, after the derivation, the 4 turns up in the bottom. Sooo, by
observation

Let's try another one. Suppose 76 exponentially decays to 31 in 5 days.

t/5
The equation is N = 76 (31/76) . Simple, isn't it?

Example 2—

Radioactive strontium 90 exponentially decays. Its half-life is 28 years. After an atomic attack, strontium 90
enters all higher life and is not safe until it decreases by a factor of 1000. How many years will it take strontium
90 to decay to safe levels after an atomic attack?

The equation, the short way, is S = So (½) t/28 . The ½ is for the half-life, or half the amount of radioactivity.

We can let So = 1000 and S = 1 for a reduction factor of 1000.

ln (.001) = (t/28) In (.5). t = 28 In (.001)/ln (.5) = 279 years to be safe.

We must truly be careful not to unleash nuclear bombs!!

Interest is also an exponential function. Simple interest = principle times rate times time. If t = 1 yr, i = pr and
the total amount A = p + pr = p(1 + r). In other words, after 1 year, the principle is multiplied by 1 = pr. After 2
2
years? A = p (1 + r) . After t years? A = p (1 + r) .
t
Suppose we have compounding interest twice a year, or half the interest rate (r/2) but twice as many periods
(2t). A = p (1 + r/2) . Compounded n times a year, the formula is A = p (1 + r/n) .
nt
2t
Finally, if the interest is compounded continuously, , and A = p e .
rt
Note 1