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MAT 141
Online
Supplemental Instruction & Conceptual Activity
Chapter 8 – Doubling Time & Half-‐Life
Ellingson
Doubling Time & Exponential Growth.
Sometimes you will be given the rate of growth instead of the doubling time. Use
this formula to approximate the doubling time.
Example: What is the approximate doubling time of a population that is increasing
exponentially at a rate of 5% per year?
Approximate Doubling Time = 70/5 = 14 years
Once you have the doubling time, you can use the exponential model here to find
predict future values.
Example: If the population from above (14 years doubling time) begins with “x”
amount of people in 2010, what will the population be in 2020. (Hint: Please do not
do this by guessing with your doubling time, actually plug the numbers in. It’s easy.
And it’s required.)
New Value: V
Initial Value: x (you may or may not be given a number here to plug in)
t: time in years since 2010 = 10 years
Tdouble: doubling time = 14 years
V = (x)(210/14)
V = 1.64x
Prediction: The population will grow by a factor of about 1.64 by
2020.
MAT 141
Online
Supplemental Instruction & Conceptual Activity
Chapter 8 – Doubling Time & Half-‐Life
Ellingson
Half-life and Exponential Decay. The half-‐life approximation works the same way.
If you aren’t given the actual half-‐life, but you have the rate of decay, then you can
use this approximation formula to find a good estimate of the half-‐life to work with.
Example: What is the half-‐life of a mineral that decays at a rate of 8% per year?
Approximate Half-‐life = 70/8 = 8.75 years
*Please note that the unit may not always be in years. You should pay attention to
the units and always label appropriately.
Just like with doubling time, once you have your half-‐life, you can predict future
values using this decay model.
Example: According to Wikipedia, the half-‐life for ibuprofen is 1.8-‐2 hours. We will
use 1.8 to make things more interesting (because everyone loves decimals ). The
maximum dosage for adults is 800 mg. If you take 800 mg of ibuprofen , how much
is still in your system right before you take it again 6 hours later?
New Value: V
Initial Value: 800 mg
t: 6 hours
Thalf: 1.8 hours
V = 800 (1/2)6/1.8 (Remember order of operations!)
V = 79.37
Prediction: You will have approximately 79.37 mg of ibuprofen left in
your body 6 hours after taking the medicine.
MAT 141
Online
Supplemental Instruction & Conceptual Activity
Chapter 8 – Doubling Time & Half-‐Life
Ellingson
Conceptual Activity. Complete the following exercises and submit your answers in
the appropriate form in Canvas.
1. The number of cells in a tumor doubles ever 6 months. If the tumor begins
with a single cell, what is the exponential model that shows how many cells,
C(t), there are in the tumor after a given number of years, t? (note that the
doubling time must be converted to years and you may use the ^ symbol to
indicate something is being raised to a power)
2. How many cells will there be after 6 years? You must show use of the formula
to get credit for this problem. Do not just double it by hand 12 times. Show
your work and answer the problem in a complete sentence.
3. A community of rabbits begins with an initial population of 100 and grows
7% per month. Find the approximate doubling time for the rabbit population.
Label appropriately.
4. What is the exponential model for the growth of the rabbit population. Let
R(m) be the total population and let m be the time, in months since the initial
tracking of the population.
5. How many rabbits will there be after a year and a half? Show how you use
the formula, and use a complete sentence to answer the question. Use logic to
determine how you should round the number.
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