# Thread: Half life problem

1. ## Half life problem

Here is the question:
An unknown radioactive element decays into non-radioactive substances. In days the radioactivity of a sample decreases by percent.

(a) What is the half-life of the element?
(b)How long will it take for a sample of mg to decay to mg?

I've tried it several times and I keep getting the same answers which are wrong

I used $dy/dx=-ky$ I integrate this and got $y=ce^{-kt}$.

I used $y(0)=y_0$ which gave me the equation $y=y_0e^{-kt}$.

I then plugged in $0.37y_0$ for y and 400 for t from the question. this gave me k= 0.0025. giving me the equation $y(t)=y_0e^{-.0025t}$.

I then set $y(t)= 0.5y_0$ and solved for t to get the half life. Which gave me the answer 277 days.

What am I doing wrong?

2. 1) Why do you think it is wrong?

2) Check your notation. How do you get from dy/dx = -ky to y = f(t)? It's okay, since everyone knows what you mean, but you should write it as clearly as you can.

3) You rounded 'k' rather severely. Use more decimal places.

3. Originally Posted by cmerickson21
Here is the question:
An unknown radioactive element decays into non-radioactive substances. In days the radioactivity of a sample decreases by percent.

(a) What is the half-life of the element?
(b)How long will it take for a sample of mg to decay to mg?

I've tried it several times and I keep getting the same answers which are wrong

I used $dy/dx=-ky$ I integrate this and got $y=ce^{-kt}$.

I used $y(0)=y_0$ which gave me the equation $y=y_0e^{-kt}$.

I then plugged in $0.37y_0$ for y and 400 for t from the question. this gave me k= 0.0025. giving me the equation $y(t)=y_0e^{-.0025t}$.

I then set $y(t)= 0.5y_0$ and solved for t to get the half life. Which gave me the answer 277 days.

What am I doing wrong?
I always find this approach to half life problems bizarrely over complex.

If half-life means anything it tells you that if you start at $t=0$ with an initial quantity $m_0$ which decays (exponentially) with half-life $t_{1/2}$, then the quantity remaining at $t$ is:

$m=m_0 2^{-\frac{t}{t_{1/2}}}$

CB