# Math Insight

### Doubling time and half-life of exponential growth and decay

Math 1241, Fall 2020
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Due date: Sept. 25, 2020, 11:59 p.m.
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Total points: 3
1. A population of sea lions is decreasing at a rate of 5% per year.
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.)

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.)

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.)

2. 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.

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?

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
:

.

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?
:

3. 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.

4. 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.)

5. 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?