SOLUTION: Below is a list of elements and their known half-life. Using the given half life determine an exponential function that gives the proper decay rate for the given element. Create a

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Below is a list of elements and their known half-life. Using the given half life determine an exponential function that gives the proper decay rate for the given element. Create a       Log On


   

Question 1119205: Below is a list of elements and their known half-life. Using the given half life determine an exponential function that gives the proper decay rate for the given element. Create a table of values listing the amount of the element you started with and then determining the amount of time to go from 1,000 units to the amount in the table
Amount Remaining Years/Days to Decay to Remaining Amount 1000 900 800 700 600 500 (half life) 400 300 200 100 0

Make sure to show the work for each calculation as a function of time, A(t), where A is the amount remaining and t is time in years/days (depending on the half-life value.) You will be using a log function to calculate the time.

Make a graph of your table plotting out all points in the table. Use an appropriate scale for your graph and state the scale. Your axes should reflect the function A(t) (meaning one should be labeled A and the other t.)

Element/Isotope Half-Life Barium (Ba) 133 10.5 years
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
Below is a list of elements and their known half-life.
Using the given half life determine an exponential function that gives the proper decay rate for the given element.
Create a table of values listing the amount of the element you started with and then determining the amount of time to go from 1,000 units to the amount in the table.
Amount Remaining Years/Days to Decay to Remaining Amount 1000 900 800 700 600 500 (half life) 400 300 200 100 0
:
Make sure to show the work for each calculation as a function of time, A(t), where A is the amount remaining and t is time in years/days (depending on the half-life value.) You will be using a log function to calculate the time.
:
Using the radioactive decay equation A = Ao*2^(-t/h), where
A = amt after t time
Ao = initial amt
t = decay time (yrs)
h = half-life of substance
Therefore
1000*2^(-t/10.5) = A
2^(-t/10.5) = A%2F1000
we want to find t in terms of A
Using natural logs
Ln(2^(-t/10.5) = ln%28A%2F1000%29
-t%2F10.5ln(2) = ln%28A%2F1000%29
t = ln%28A%2F1000%29%2Fln%282%29 * -10.5
We can substitute for A and find t, for example for remaining value of 900
t = ln%28900%2F1000%29%2Fln%282%29 * -10.5
do the math and we have
t = 1.6 yrs
Do this for all the other values, your table should be:
A | t
------
900| 1.6
800| 3.4
700| 5.4
600|
500|10.5
400|
300|
200|
100|34.9
I'll let you calculate the missing times,
:
Make a graph of your table plotting out all points in the table. Use an appropriate scale for your graph and state the scale. Your axes should reflect the function A(t) (meaning one should be labeled A and the other t.)
Graph the original equation; y = 1000*2^(-x/10.5)
+graph%28+300%2C+200%2C+-10%2C+80%2C+-200%2C+1200%2C+1000%2A2%5E%28-x%2F10.5%29%2C+500%29+
time on the horizontal and A(t) on the vertical. green = 10.5, 500
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Element/Isotope Half-Life Barium (Ba) 133 10.5 years
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