The burial cloth of an Egyptian mummy is estimated to contain 560 g of the radioactive materialcarbon-14, which has a half life of 5730 years. a. Complete the table below. Make sure you justify your answer by showing all the steps. begin{array}{|l|l|l|}hline t(text{in years})&m(text{amoun of radioactive material})hline0&hline5730hline11460hline17190hlineend{array} b. Find an exponential function that models the amount of carbon-14 in the cloth, y, after t years. Make sure you justify your answer by showing all the steps. c. If the burial cloth is estimated to contain 49.5% of the original amount of carbon-14, how long ago was the mummy buried. Give exact answer. Make sure you justify your answer by showing all the steps.

Question
Exponential models
asked 2021-02-12
The burial cloth of an Egyptian mummy is estimated to contain 560 g of the radioactive materialcarbon-14, which has a half life of 5730 years.
a. Complete the table below. Make sure you justify your answer by showing all the steps.
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{l}\right|}{l}{\left|{l}\right|}\right\rbrace}{h}{l}\in{e}{t}{\left(\text{in years}\right)}&{m}{\left(\text{amoun of radioactive material}\right)}\backslash{h}{l}\in{e}{0}&\backslash{h}{l}\in{e}{5730}\backslash{h}{l}\in{e}{11460}\backslash{h}{l}\in{e}{17190}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
b. Find an exponential function that models the amount of carbon-14 in the cloth, y, after t years. Make sure you justify your answer by showing all the steps.
c. If the burial cloth is estimated to contain 49.5% of the original amount of carbon-14, how long ago was the mummy buried. Give exact answer. Make sure you justify your answer by showing all the steps.

Answers (1)

2021-02-13
Given that
Initially the radioactive carbon-14 present in the burial cloth of the egyptian mummy is 560g.
The half life of carbon-14 is 5730 years.
The total amount of carbon-14 decayed after t years is given by
\(\displaystyle{A}={A}_{{0}}{\left({0.5}\right)}^{{{\frac{{{1}}}{{{h}}}}}}\) (1)
Where, \(\displaystyle{A}_{{0}}\) initial amount of carbon-14
t-time in years,
h-Half life of carbon-14
a) To complete table using the following calculation:
Now,
1) t=0 years, h=5730 years,\(\displaystyle{A}_{{0}}={560}\)g
\(\displaystyle{A}={A}_{{0}}{\left({0.5}\right)}^{{{\frac{{{1}}}{{{h}}}}}}\)
\(\displaystyle{A}={560}{\left({0.5}\right)}^{{{\frac{{{0}}}{{{5730}}}}}}\)
\(\displaystyle{A}={560}{\left({0.5}\right)}^{{0}}={560}{\left({1}\right)}\)
\(\displaystyle{A}={560}\)g
2) t=5730 years, h=5730 years,\(\displaystyle{A}_{{0}}={560}\)g
\(\displaystyle{A}={560}{\left({0.5}\right)}^{{{\frac{{{5730}}}{{{5730}}}}}}\)
\(\displaystyle{A}={560}{\left({0.5}\right)}^{{1}}={560}{\left({0.5}\right)}\)
\(\displaystyle{A}={280}\)g
3) t=11460 years, h=5730 years,\(\displaystyle{A}_{{0}}={560}\)g
\(\displaystyle{A}={560}{\left({0.5}\right)}^{{{\frac{{{11460}}}{{{5730}}}}}}\)
\(\displaystyle{560}{\left({0.5}\right)}^{{2}}={560}{\left({0.25}\right)}\)
\(\displaystyle{A}={140}\)g
3) t=17190 years, h=5730 years,\(\displaystyle{A}_{{0}}={560}\)g
\(\displaystyle{A}={560}{\left({0.5}\right)}^{{{\frac{{{17190}}}{{{5730}}}}}}\)
\(\displaystyle{A}={560}{\left({0.5}\right)}^{{3}}={560}{\left({0.125}\right)}\)
\(\displaystyle{A}={70}\)g
The table becomes
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{l}\right|}{l}{\left|{l}\right|}\right\rbrace}{h}{l}\in{e}{t}{\left(\text{in years}\right)}&{m}{\left(\text{amoun of radioactive material}\right)}\backslash{h}{l}\in{e}{0}&{560}\backslash{h}{l}\in{e}{5730}&{280}\backslash{h}{l}\in{e}{11460}&{140}\backslash{h}{l}\in{e}{17190}&{70}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
b)To find the exponential function that models the amount of carbon-14 in the cloth, y, after t years:
\(\displaystyle{A}={A}_{{0}}{e}^{{-{k}{t}}}\)
Here, At t=0, y=560 g,
k- rate of decay,
t- time in years.
The exponential function is
\(\displaystyle{y}={560}{e}^{{-{k}{t}}}\)
c) To find t, for k=49.5% :
49.5% of original amount of carbon-14 is
\(\displaystyle{49.5}\%\times{560}={\frac{{{49.5}}}{{{100}}}}\times{560}\)
\(\displaystyle={49.5}\times{5.6}\)
\(\displaystyle{49.5}\%\times{560}={277.2}\)g
Now, A0 =560, A= 277.2 g, h=5730 years, from (1)
\(\displaystyle{A}={A}_{{0}}{\left({0.5}\right)}^{{{\frac{{{t}}}{{{h}}}}}}\)
\(\displaystyle{277.2}={560}{\left({0.5}\right)}^{{{\frac{{{t}}}{{{5730}}}}}}\)
\(\displaystyle{\left({0.5}\right)}^{{{\frac{{{t}}}{{{5730}}}}}}={\frac{{{277.2}}}{{{560}}}}\)
\(\displaystyle{\left({0.5}\right)}^{{{\frac{{{t}}}{{{5730}}}}}}={0.495}\)
\(\displaystyle{\frac{{{t}}}{{{5730}}}}{\ln{{0.5}}}={\ln{{0.495}}}\)
\(\displaystyle{t}={5730}{\left({\frac{{{\ln{{0.495}}}}}{{{\ln{{0.5}}}}}}\right)}\)
\(\displaystyle{t}={5730}{\left({\frac{{-{0.7032}}}{{-{0.693}}}}\right)}\)
\(\displaystyle{t}={5730}{\left({1.015}\right)}\)
\(\displaystyle{t}={5814.33}\) years
Thus, the mummy was burried before 5814.33 years.
0

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The burial cloth of an Egyptian mummy is estimated to contain 560 g of the radioactive materialcarbon-14, which has a half life of 5730 years.
a. Complete the table below. Make sure you justify your answer by showing all the steps.
\(\begin{array}{|l|l|l|}\hline t(\text{in years})&m(\text{amoun of radioactive material})\\\hline0&\\\hline5730\\\hline11460\\\hline17190\\\hline\end{array}\)
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The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
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Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
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Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P \((A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).\)
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
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