Question

The half - life of a certain radioactive material is 85 days. An initial amount of the material has a mass of 801 kg Write an exponential function that models the decay of this material. Find how much radioactive material remains after 10 days. Round your answer to the nearest thousandth.

Exponential models
ANSWERED
asked 2021-01-02
The half - life of a certain radioactive material is 85 days. An initial amount of the material has a mass of 801 kg Write an exponential function that models the decay of this material. Find how much radioactive material remains after 10 days. Round your answer to the nearest thousandth.

Answers (1)

2021-01-03
Given initial mass:
\(\displaystyle{m}_{{0}}={801}\)kg at t=0
Let
m(t)=Mass at time t
The exponential function,
\(\displaystyle{m}{\left({t}\right)}={m}_{{0}}{e}^{{{k}{t}}}\), where k =constant
Also given:
half life=85days
\(\displaystyle\Rightarrow{m}{\left({85}\right)}={\frac{{{m}_{{0}}}}{{{2}}}}\)
Now,
When t=85days:
\(\displaystyle{m}{\left({85}\right)}={m}_{{0}}{e}^{{{k}\times{85}}}\)
\(\displaystyle\Rightarrow{\frac{{{m}_{{0}}}}{{{2}}}}={m}_{{0}}{e}^{{{85}{k}}}\)
\(\displaystyle\Rightarrow{e}^{{{85}{k}}}={\frac{{{1}}}{{{2}}}}\)
\(\displaystyle\Rightarrow{e}^{{k}}={\left({\frac{{{1}}}{{{2}}}}\right)}^{{{\frac{{{1}}}{{{85}}}}}}\)
\(\displaystyle\Rightarrow{k}={{\ln{{\left({\frac{{{1}}}{{{2}}}}\right)}}}^{{{\frac{{{1}}}{{{85}}}}}}}\)
\(\displaystyle\Rightarrow{k}=-{\frac{{{1}}}{{{85}}}}{\ln{{2}}}\)
Therefore,
When t=10days:
\(\displaystyle{m}{\left({10}\right)}={801}{e}^{{{k}\times{10}}}\)
\(\displaystyle={801}{\left[{e}^{{{10}\times{\left(-{\frac{{{1}}}{{{85}}}}{\ln{{2}}}\right\rbrace}}}\right]}\)
\(\displaystyle={738.273}\)kg
Hence,
The radioactive material remains after 10 days =738.273kg
0
 
Best answer

expert advice

Need a better answer?

Relevant Questions

asked 2021-05-26
You open a bank account to save for college and deposit $400 in the account. Each year, the balance in your account will increase \(5\%\). a. Write a function that models your annual balance. b. What will be the total amount in your account after 7 yr? Use the exponential function and extend the table to answer part b.
asked 2020-11-22
Scientists are working with a sample of cobalt-56 in their laboratory. They begin with a sample that has 60 mg of cobalt-56, and they measure that after 31 days, the mass of cobalt-56 sample is 45.43 mg. Recall that the differential equation which models exponential decay is \(\displaystyle{\frac{{{d}{m}}}{{{\left.{d}{t}\right.}}}}=-{k}{m}\) and the solution of that differential equation if \(\displaystyle{m}{\left({t}\right)}={m}_{{0}}{e}^{{-{k}{t}}}\), where \(\displaystyle{m}_{{0}}\) is the initial mass and k is the relative decay rate.
a) Use the information provided to compute the relative decay rate k. Show your calculation (do not just cit a formula).
b) Use the information provided to determine the half-life of cobalt-56. Give your answer in days and round to the second decimal place. Show your calculation (do not just cite a formula).
c) To the nearest day, how many days will it take for the initial sample of 60mg of cobalt-56 to decay to just 10mg of cobalt-56?
d) What will be the rate at which the mass is decaying when the sample has 50mg of cobalt-56? Make sure to indicate the appropriate units and round your answer to three decimal places.
asked 2021-05-16
A computer valued at $1500 loses 20% of its value each year. a. Write a function rule that models the value of the computer. b. Find the value of the computer after 3 yr. c. In how many years will the value of the computer be less than $500? Use the exponential function to answer part b.
asked 2021-05-07
A share of ABC stock was worth $60 in 2005 and only worth $45 in 2010. a. Find the multiplier and the percent decrease. b. Write an exponential function that models the value of the stock starting from 2005. c. Assuming that the decline in value continues at the same rate , use your answer to (b) to predict the value in 2020.
asked 2021-05-12
When a < 0 and b>1,y=abx models negative exponential growth. a. Write an exponential function that models negative growth. b. Give an example of a situation that could be modeled by your function. c. Explain one difference between negative exponential growth and exponential decay.
asked 2021-05-04
You decide to start saving pennies according to the following pattern. You save 1 penny the first day, 2 pennies the second day, 4 the third day, 8 the fourth day, and so on. Each day you save twice the number of pennies you saved on the previous day. Write an exponential function that models this problem. How many pennies do you save on the thirtieth day?
asked 2021-05-25
Radioactive substances decay exponentially. For example, a sample of Carbon\(\displaystyle-{14}{\left(^{\left\lbrace{14}\right\rbrace}{C}\right)}\)will lose half of its mass every 5730 years. (In other words, the half-life of \(\displaystyle^{\left\lbrace{14}\right\rbrace}{C}\) is 5730 years.) Let A be the initial mass of the sample. Model the decay of \(\displaystyle^{\left\lbrace{14}\right\rbrace}{C}\)using a discrete-time model... (a) using \(\displaystyle\delta{t}={5730}\) years. (b) using Δt=1year.
asked 2021-05-13
The table shows the number of employees at a software company in various years.
Years Since 2000 4 6 8 10 Number of Employees 32 40 75 124
Use your calculator to find an exponential equation that models the growth of the company.
asked 2021-05-13
The population of Williston, North Dakota, has grown rapidly over the past decade due to an oil boom. The table gives the population of the town in 2007, 2009, and 2011. Years Since 20007811 Population (thousands)12.413.016.0
Use your calculator to find an exponential equation that models the growth of the town.
asked 2021-06-09
With what kind of exponential model would half-life be associated? What role does half-life play in these models?
...