Solving Radioactive Decay Equations: A Step-by-Step Guide for Beginners

[x0hkatielee]

I've just been really thrown off by what this problem is asking me.

Given:
The decay of a radioactive material may be modeled by assuming that the amount A(t) of material present (in grams) at time t (minutes) decays at a rate proportional to the amount present, that is dA/dt= -kA for some positive constant k. Every subpart of this question refers to exactly the same radioactive material.

Question:
a. derive an equation for the amount A(t) present at time t in terms of the constant k and the amount A(o) present at time t=0

b. if A(5) = 1/3A(3), find K

c. at what time t will the amount A(t) be 1/4A(0)

My attempt (?):
for part a I wasnt sure if it was referring to just giving the equation A(t)=A(0)e^-kt
and I have no idea how to go about b or c. please help me if possible ! I think I'm over thinking the problem. I'm just really thrown off by the lack of numbers. Thank you!

 

[rock.freak667]

For part a, you were given dA/dt = -kA, you need to solve this first order DE to derive the equation you want. Are you able to do this?

For part b), once you have the formula from part a, you just need to substitute the numbers.

For example A(1) means the value of A when t=1.

 

[Mark44]

I've just been really thrown off by what this problem is asking me.

Given:
The decay of a radioactive material may be modeled by assuming that the amount A(t) of material present (in grams) at time t (minutes) decays at a rate proportional to the amount present, that is dA/dt= -kA for some positive constant k. Every subpart of this question refers to exactly the same radioactive material.

Question:
a. derive an equation for the amount A(t) present at time t in terms of the constant k and the amount A(o) present at time t=0

b. if A(5) = 1/3A(3), find K

c. at what time t will the amount A(t) be 1/4A(0)

My attempt (?):
for part a I wasnt sure if it was referring to just giving the equation A(t)=A(0)e^-kt

This is fine, except you need some parentheses for the exponent.
A(t)=A(0)e^(-kt)

and I have no idea how to go about b or c.

What is A(5)? What is A(3)?
Substitute these expressions into the the equation A(5) = (1/3)A(3) and solve for k.

For the last one, you should have gotten the value of k in part b, so you should be able to do it by then.

please help me if possible ! I think I'm over thinking the problem. I'm just really thrown off by the lack of numbers. Thank you!

 

[x0hkatielee]

For part a, you were given dA/dt = -kA, you need to solve this first order DE to derive the equation you want. Are you able to do this?

For part b), once you have the formula from part a, you just need to substitute the numbers.

For example A(1) means the value of A when t=1.

Thanks for the reply! I'm a little rusty at calc because I haven't taken it in a while. I'm not sure exactly how to get the formula from the equation given.

 

[x0hkatielee]

This is fine, except you need some parentheses for the exponent.
A(t)=A(0)e^(-kt)

What is A(5)? What is A(3)?
Substitute these expressions into the the equation A(5) = (1/3)A(3) and solve for k.

For the last one, you should have gotten the value of k in part b, so you should be able to do it by then.

Thanks for the reply! What is throwing me off is not having numbers or any values for A(3) or A(5).

 

[Mark44]

For part a, it appears that you need to derive the equation you showed, not just merely write it down. This entails solving the diff. eqn. dA/dt = -kA.

This equation is fairly simple to solve, using separation of variables, one of the first techniques that you learn in diff. equations.

dA/dt = -kA => dA/A = -k dt

Part b assumes that you have a formula (function) for A(t). Use the formula to find A(5) and A(3), and substitute these values into the equation A(5) = (1/3) A(3).

 

[x0hkatielee]

I got the equation the way you said to, but the problem doesn't give me any values for A(0), A(3), or A(5). And this is really confusing me. This problem has been stumping me for a long time because it doesn't give me much to work with. Sorry if I'm bugging you!

 

[Mark44]

I got the equation the way you said to, but the problem doesn't give me any values for A(0), A(3), or A(5).

It doesn't need to. Show us what you did.

And this is really confusing me. This problem has been stumping me for a long time because it doesn't give me much to work with. Sorry if I'm bugging you!

It gives you all the information you need.

 

[x0hkatielee]

I had dA/dt=-kA and rearranged it and integrated both sides to get to the equation A(t)=A(0)e^(-kt). now I know I'm supposed to plug the values for A(5) and A(3) into part b to solve for k, but how do I get those values if I don't know A(0) or k.

 

[Mark44]

What is A(5)?
What is A(3)?

 

[x0hkatielee]

the only thing that they give me towards figuring that out is that A(5)=(1/3)A(3)... ? Not sure if that's what you meant. I wouldn't be able to solve for them using the equation for part a without knowing K or A(0).

 

[Mark44]

the only thing that they give me towards figuring that out is that A(5)=(1/3)A(3)... ? Not sure if that's what you meant.

That's not what I asked, and it's not what I meant.

You have a formula for A(t). Evaluate your formula at t = 5 and t = 3. What do you get?

I wouldn't be able to solve for them using the equation for part a without knowing K or A(0).

Both expressions (A(5) and A(3)) will have k and A(0) in them. You don't need to know these values to write A(5) and A(3), though.

 

[x0hkatielee]

Okay so I think I understand ! If I set A(5)=(1/3)A(3) in the equations then the A(0) values cancel and I can solve for k=(ln 3)/2.. hope that's right!

 

[LCKurtz]

My question this whole time has been how to get those two values. I get what you're saying, but I really don't understand how I can plug them into the formula A(t)=A(0)e^(-kt) without knowing more. I could put A(3)=A(0)e^(-k(3)) and do the same for A(5) but that doesn't help me.. at least I don't think so. :(

Of course it will help. Once you have A(5) too, maybe you can use the given fact that A(5) = (1/3)A(3) to solve for k. Did you try that?

 

[x0hkatielee]

Of course it will help. Once you have A(5) too, maybe you can use the given fact that A(5) = (1/3)A(3) to solve for k. Did you try that?

I think I finally got it right. so like A(0)e^-(5k)=(1/3)A(0)e^-(3k)... and the A(0)'s cancel out. So I get e^-(5k)=(1/3)e^-(3k) . and then once I simplify more I can find k

 

[x0hkatielee]

okay so it looks like what I'm getting for part b is k= (ln 3)/-2 ...? Is this right or am I completely off.

 

[Mark44]

You have a mistake, but you're on the right track.

 

[x0hkatielee]

it's the negative sign, right? I noticed that.

 

[Mark44]

Show how you got k...