8B Doubling
Time and Half Time
Review:
A logarithm is a number which represents
a power or exponent:
log10 x is the power to which 10 must be raised to get
x.
log10
10,000 = 4,
log10
0.0001 = - 4
some log rules:
log10 10x =
x
log 10(x * y) = log10
x + log10 y
log10 ax = x * log10
a
do without calculator, explain why / why not:
-
100.928 is between 10 and 100.
-
10-5.2 is between 100,000 and 1,000,000.
-
log10 1,600,000 is between 6 and 7.
-
log10 (8 * 10 ) is between 12 and 13
-
log10 1/4 is between 0 and 1.
given that log 5 = 0.699, find without calculator
-
log10 50
-
log10 25
-
log10 0.20
The doubling time is the time required
for each doubling of a quantity in exponential growth.
some bacteria double in number every hour...
Calculations with the doubling time:
new value = initial value * 2t / Tdouble
where
Tdouble = doubling time
t = time elapsed since the initial value at time t = 0.
-
The doubling time of a population of flies is 3 hours. By what factor does
the population increase in 24 hours? in 1 week?
-
The number of cells in a tumor doubles every 1.5 months. If the tumor begins
as a single cell, how many cells are there after 3 years? after 4 years?
Approximate Doubling Time Formula ( Rule of 70):
For a quantity growing exponentially at a rtate of P %
per time period, the doubling time is aproximately:
Tdouble =roughly70
/ P
estimate is good only when P < 15, P small.
-
The Cinsumer Price Index is increasing at a rate of 7 % per year.
What is its doubling time? By what factor will prices increase in 3 years?
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Oil consumption is increasing at a rate of 1.9 % per year. What is its
doubling time? By what factor will oil consumption increase in a decade?
Exponential Decay: a quantity decreases
by the same percentage every fixed time period.
e.g. y = 1000 * (2 /3)t, where t is time
half life: time elapsed between starting time and when quantity has decreased
to half its original size.
Plutonium, Pu-239, has a half life of 24,000 years.
Calculations with the half life:
new value = initial value * (1 / 2)t / Thalf
where Thalf = half life,
t = time elapsed
since start.
-
The half life of a drug in the blood stream is 12 hours. By what factor
does the concentration of the drug decrease in 24 hours? in 36 hours?
-
Radium-226 is a metal with a half life of 1600 years. If you start
with 1 kilogram of radium-226, how much will remain after 1000 years?
after 10,000 years?
Approximate Half Life:
For a quantity decaying exponentially at a rate of P % per time
period, the half life is approximately:
Thalf =roughly70
/ P
estimate works only if P < 15, P small.
-
Urban encroachment is causing an area of a forest to decline at a rate
of 10 % per year. What is the half life of the forest? What fraction of
the forest will remain in 100 years?
Exact doubling time and half life formulas
Suppose r is the growth rate, as a decimal.
When the growth rate is positive ( > 0), then we deal with exponential
growth:
Tdouble = log10 2 / log10 (1
+ r)
when the growth rate is negative ( < 0), then we deal with exponential
decay:
Thalf = - log10 2 / log10
(1 + r)
-
Hyperinflation is driving up prices at a rate of 80 % per month. For an
item that costs $1000 today, what will the price be ina year?
-
A nation of 100 million people is growing at a rate of 3 % per year. What
will its population be in 50 years?
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