Graphing Radiation Decay

[Bluebottle]

Hi Everyone:

I am a volunteer science tutor, so not really a math tutor, but sometimes a math problem comes up, but I am not having any luck figuring this one out on my own:

Help would be appreciated!

Here is the equation:

M=M_o (0.5)ⁿ

M=initial isotope mass
M_o=final isotope mass
n=number of half lives (time)

Q: Start with 100 grams and some time later you have 35 grams left. How many half lives have gone by? Graph the solution.

My Answer: 100=35(0.5)ⁿ solve for n
n=log35/log(0.5)
n= -5.13

n(time) can't be negative so I know its wrong

I haven't done log equations for years so not sure where I'm messing up.

Thanks for your help

 

[MarkFL]

I think the issue may lie in the fact that \(M\) is the present mass and \(M_0\) is the initial mass. So, we'd have:

[MATH]35=100\left(\frac{1}{2}\right)^n[/MATH]
[MATH]\frac{7}{20}=\left(\frac{1}{2}\right)^n[/MATH]
[MATH]\frac{20}{7}=2^n[/MATH]
[MATH]n=\frac{\ln\left(\frac{20}{7}\right)}{\ln(2)}\approx1.514573172829758[/MATH]
This answer makes sense, because after one half-life there is 50 g, and after 2 half-lives there is 25 g, and so we know the answer must be between 1 and 2.

 

[Bluebottle]

Thanks very much Mark, my mistake.

So to graph it I would just make a table of values for "n" (different half lives)?

Thanks again

 

[MarkFL]

I would plot a graph like this:



The curve in green is the function:

[MATH]f(x)=100\cdot2^{-n}[/MATH]
This represents the amount of isotope mass (in grams) present at time \(n\), where \(n\) represents the half-life of the isotope. I have shown points of the curve for integer values of \(n\). The horizontal red line is \(y=35\), and the first coordinate of the point of intersection is the number of half-lives it takes for the mass to decay to 35 grams.

 

[Bluebottle]

I would plot a graph like this:

View attachment 16424

The curve in green is the function:

[MATH]f(x)=100\cdot2^{-n}[/MATH]
This represents the amount of isotope mass (in grams) present at time \(n\), where \(n\) represents the half-life of the isotope. I have shown points of the curve for integer values of \(n\). The horizontal red line is \(y=35\), and the first coordinate of the point of intersection is the number of half-lives it takes for the mass to decay to 35 grams.

Hi Mark:

Thank you so much, you have been very helpful. I will share this with my student (it's a First Nation Adult Education School) and we will study it together and use it to help us move ahead in the unit.

 

[Steven G]

My Answer: 100=35(0.5)ⁿ solve for n
n=log35/log(0.5)
n= -5.13

What happened to the 100??????