Doubling time and half-life of exponential growth and decay

Math 201, Spring 2017
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A population of sea lions is decreasing at a rate of 5% per year.
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- Doubling time and half-life of exponential growth and decay
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1. If the population continues to decline at this rate, to what fraction of the original population size will the population decline after 10 years?
  (Keep at least 4 significant digits.)
  
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  Hint
  
  If the population is decreasing at a rate of 5% per year, then by what number do we need to multiply the population size each year?
  
  We didn't tell you how many sea lions there were to start out with, so apparently, your answer shouldn't depend on that value. If you like, you can define a variable to represent the initial population size (for example $p_0$); then, you can multiply that initial population size by a certain number once for each year that passes. However, since we are asking for what fraction of the original population size is left (for example, what fraction of the original $p_0$), you need to divide by that original population size to get your answer. In the end, the variable you chose for initial population size should drop out.
  
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2. To what fraction of the original population size will the population decline after $n$ years?
  
  (Online, enter exponentiation using ^, so enter $a^b$ as a^b.)
  
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  Hint
  Same procedure as for the previous part, except that you have multiply by that number $n$ times.
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3. To find how long it will take for the population to decline by one-half, follow these steps. Set the expression from part (b) equal to one-half.
  
  $ = \frac{1}{2}$
  
  Take the logarithm of both sides of that equation.
  
  =
  
  (Online, you can use either ln or log for logarithm; both are interpreted a logarithm base $e$. In this case, it doesn't matter what base logarithm you use. Also, don't simplify your answers yet, but leave them as the logarithm of your previous answers.)
  
  Using the log of power rule, bring down the exponent from the left hand side in front of the logarithm.
  
  =
  
  Solve for the value of $n$. The result is a ratio of logarithms.
  
  $n=$
  $\approx$
  
  (In the first blank, write the ratio of logarithms. In the second blank, give a decimal approximation with at least 4 significant digits.)
  
  To repeat, how long will it take for the population to decline by one-half?
  
  (The second blank is for a unit.)
4. You can use a similar procedure to find out how long it will take for the population to decline to one-tenth its original size. Set the expression from part (b) equal to one-tenth.
  $ = \frac{1}{10}$
  
  Take the logarithm of both sides of that equation.
  
  =
  
  (Online, you can use either ln or log for logarithm; both are interpreted a logarithm base $e$. In this case, it doesn't matter what base logarithm you use.)
  
  Using the log of power rule, bring down the exponent from the left hand side in front of the logarithm.
  
  =
  
  Solve for the value of $n$. The result is a ratio of logarithms.
  $n=$
  $\approx$
  
  (In the first blank, write the ratio of logarithms. In the second blank, give a decimal approximation with at least 4 significant digits.)
  
  To repeat, how long will it take for the population to decline to one-tenth of its original size?
  
  (The second blank is for a unit.)
5. If the current population size is 100,000, how long will it take for the population drop down to 5,000 sea lions?
  
  (Keep at least 4 significant digits. Second blank is for a unit.)
  
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  Hint
  To get to 5,000 sea lions, you need to get down to what fraction of the original population? The rest is the same as the previous problems.
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Bacteria are growing in a beaker so that the population size increases by 14.87% every minute.
1. If $b_t$ is the bacteria population size in minute $t$, set up a dynamics system model that describes the evolution of the population size.
  $b_{t+1} - b_t =$
  , for $t=0,1,2,3 \ldots$
2. How long does it take the population size to double?
  
  $T_{\text{double}} = $
  
  Your answer will look a lot prettier if you round to four significant digits. (In this case, this means round to the nearest thousandth, as there should be one digit to the left of the decimal.) The second blank is for a unit.
  
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  Hint
  Hint: first rewrite the dynamical system into function iteration form. Then, solve the system for $b_t$. Then, find the condition for doubling. Alternatively, use the formula for doubling time.
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3. If the population continues to grow at this rate, by what factor does the population size increase in one hour?
  In two hours?
  In four hours?
  
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  Hint
  You have two ways to solve this. One way is to go back to the solution of the dynamical system and plug in the number of minutes elapsed for each interval. A different way is to calculate how many times the population size doubled in each interval and use that to infer by what factor the population size increased.
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4. An experiment is begun at midnight with just a few bacteria so that the fraction of the beaker that the bacteria occupy is approximately $0.00000005959 = 5.959 \times 10^{-8}$. With this initial condition, the bacteria completely fill the beaker after two hours, at 2 AM. At what time was the beaker half full? The beaker was half full at
  :
  
  .
  
  Write your answer in the form: hh:mm AM/PM. Round your answer to the nearest minute.
  
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  Hint
  There are two ways to answer this problem. One is the hard way, which is to start your calculation at midnight. The second way is to completely ignore the information you were given at midnight.
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5. Imagine the researchers realized before 2AM that the bacteria were about to overflow the beaker. They found three more empty beakers of the same size as the original beaker so that they had a total of four beakers to hold the bacteria. At what time did the bacteria fill all four beakers?
  :
  
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  Hint
  Thinking about the doubling time could be helpful.
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The polymerase chain reaction is a means of making multiple copies of a DNA segment from only a minute amount of original DNA. The procedure consists of a sequence of multiple cycles. During the course of one cycle, each DNA segment present is duplicated. Suppose you begin with 1 picogram = 0.000000000001 g of DNA.
If $d_n$ is the amount of DNA in grams after $n$ cycles, write a discrete dynamical system with initial condition from which the amount of DNA present at the end of each cycle can be computed.

$d_{n+1} = $
for $n=0,1,2,3,\ldots$
$d_0 =$

How many grams of DNA would be present after 30 cycles?
grams.
(Keep at least 3 significant digits in your answer.)
In the first days of life, the cells in a human embryo divide into two cells approximately every day. After fertilization, the new life consists of a single cell. If the number of cells continued to double every day, how many weeks would it take the embryo to grow to the size of a human adult, containing approximately 100 trillion ($10^{14}$) cells?
weeks
(Keep at least four significant digits in your response.)
Suppose after someone gets lead poisoning, no further lead is introduced into the bloodstream so that the amount of lead in the bloodstream decreases by 11% per week. Let $p_t$ be the amount of lead, measured in μg/dl (micrograms per deciliter), in the bloodstream $t$ weeks after the lead exposure. (See dynamical system exploration page for more on the lead decay model.)
1. If we write a dynamical system describing the lead decay in difference form, \begin{align*} p_{t+1} - p_t &= a p_t\\ p_0 &= p_0, \end{align*} what is the value of the parameter $a$? $a=$
2. If we write a dynamical system describing the lead decay in function iteration form, \begin{align*} p_{t+1} &= b p_t\\ p_0 &= p_0, \end{align*} what is the value of the parameter $b$? $b=$
3. In general, what is the relationship between $a$ and $b$? $b =$
4. If the initial lead concentration is 64 μg/dl, how long does it take to drop to 32 μg/dl?
  weeks. To 10 μg/dl (the standard elevated blood lead level for adults)?
  weeks.
  (Keep at least four significant digits in your response.)
5. If the initial lead concentration is 32 μg/dl, how long does it take to drop to 16 μg/dl?
  weeks. To 5 μg/dl (the standard elevated blood lead level for children)?
  weeks.
  (Keep at least four significant digits in your response.)
6. Does the time required for the lead to drop to half its initial concentration depend on the value of the initial lead concentration?

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Math 201, Spring 2017

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Math 201, Spring 2017