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Advanced Math - exponential equation to linear equation
Expert: Ahmed Salami - 10/31/2009
Question
Data below is of radioactive decay of a material x Time t min y Count rate N cps ( counts per second )
Time t min 10 20 30 40 50 60 70 80 90
Count Rate cps 340 236 165 112 77 53 36 25 17
It is believed the data fits a graph with an equation of the form
N=Noe-at The a should be alpha
Where No and a are constants. I have already plotted this graph.
The exponential equation can be transformed to give a linear equation lnN=lnNo-at. Prepare a table of values for plotting a straight line graph (I dont know how to do that)Plot the line and find values for No and a
Answer
Hi Stephen,
Well, the new table of values is just one that contains values of lnN and Time t(just as before). In case you're wondering, lnN refers to the natural logarithm of N and you can get all the values by using a calculator to compute lnN from N.
The graph of InN against t would be a straight line and can be used to find the values of the unknown constants.
Referring to the general equation of a straight line,
y = mx + c
where m is the slope of the line and c is the intercept on the y axis.
From your graph of lnN against t,
lnN = lnNo - αt
comparing with y = mx + c, we can see that
m = -α and c = lnNo
which ultimately means that the slope of your graph would be -α and it'll intercept the vertical axis at a value of lnNo
So now, all you need do is draw the graph and find the slope m(which i assume you can) and of course you should see the vertical intercept c. And then,
α = -m
and
No = e^c
You can always get back to me.
Regards
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