Question

algebra: concepts & connections unit 6 georgias k - 12 mathematics standards
name
will the candy last forever?
date
for her eighth birthday, shelleys grandmother gave her a full bag of candy. shelley counted her candy and found out that there were 180 pieces in the bag. as you might suspect, shelley loves candy, so she ate half the candy on the first day. then her mother told her that if she eats it at that rate, the candy will only last one more day—so shelley devised a clever plan. she will always eat half of the candy that is left in the bag each day. she thinks that this way she can eat candy every day and never run out.

1. create a table that depicts this situation. be sure to define the quantities you are using on the table.
2. based on the table from the previous question, determine a mathematical model that describes the amount of candy shelley has. define the variables you use.
3. how much candy does shelley have by the end of the week?
4. use the information from questions 1 - 3 to create a cer (claim, evidence, and reasoning) model for the following question: will the candy really last forever?

the marvel of medicine
name
date
part i
a doctor prescribes 400 milligrams of medicine to treat an infection. each hour following the initial dose, 85% of the concentration remains in the body from the previous hour.

1. complete the table showing the amount of medicine remaining after each hour.

number of hours|percent|number of milligrams remaining in the body
0|400|400
1|400(0.85)|340
2|400(0.85)(0.85)|
3|
4|
5|

1. using symbols and words, describe the functional relationship in this situation. discuss the domain and range of both the function rule and the problem situation.
2. determine the amount of medicine left in the body after 10 hours. justify your answer in two ways.
3. when does the amount of medicine still in the body reach 50 milligrams? explain how you know.
4. suppose that the level of medicine in the patients body must remain at least greater than 100 milligrams. how often does the patient need to take the medicine?

Explanation:

Step1: Create table for candy - situation

Let $d$ be the number of days and $C$ be the amount of candy left.

Number of days ($d$)	Amount of candy left ($C$)
1	$180\times\frac{1}{2}=90$
2	$90\times\frac{1}{2} = 45$
3	$45\times\frac{1}{2}=22.5$
4	$22.5\times\frac{1}{2}=11.25$
5	$11.25\times\frac{1}{2}=5.625$
6	$5.625\times\frac{1}{2}=2.8125$
7	$2.8125\times\frac{1}{2}=1.40625$

Step2: Determine mathematical model for candy

The general formula for an exponential - decay model is $C(d)=C_0\times(\frac{1}{2})^d$, where $C_0 = 180$ (the initial amount of candy) and $d$ is the number of days.

Step3: Calculate candy left at end of week

There are 7 days in a week. Substitute $d = 7$ into the formula $C(d)=180\times(\frac{1}{2})^d$.
$C(7)=180\times(\frac{1}{2})^7=180\times\frac{1}{128}=\frac{180}{128}=\frac{45}{32}=1.40625$ pieces of candy.

Step4: Create CER for candy - lasting forever

Claim: The candy will not last forever.
Evidence: The amount of candy $C(d)$ is given by the formula $C(d)=180\times(\frac{1}{2})^d$. As $d$ (the number of days) increases, the value of $C(d)$ gets smaller and smaller. For example, after 7 days, $C(7) = 1.40625$ pieces of candy.
Reasoning: In an exponential - decay function of the form $y = a\times b^x$ where $0 < b<1$ (in our case $a = 180$ and $b=\frac{1}{2}$), as $x$ (the independent variable, which is the number of days $d$ here) approaches infinity, $y$ (the amount of candy $C(d)$) approaches 0. So, eventually, the amount of candy will be so small that it will be effectively gone.

Answer:

1. See the table above for the candy - situation table.
2. The mathematical model is $C(d)=180\times(\frac{1}{2})^d$, where $C(d)$ is the amount of candy left after $d$ days and $C_0 = 180$ is the initial amount of candy.
3. $\frac{45}{32}=1.40625$ pieces of candy.
4. Claim: The candy will not last forever. Evidence: $C(d)=180\times(\frac{1}{2})^d$ and $C(d)$ gets smaller as $d$ increases. Reasoning: For an exponential - decay function $y = a\times b^x$ with $0 < b < 1$, as $x$ approaches infinity, $y$ approaches 0.