# Quick Calculus Question: Growth and Decay, Half-Life (1 Viewer)

#### xMaFF

##### Member
Good evening,

Just need help with solving 'half-life' questions for growth and decay.
There are 3 questions, ranking from 1-3 of importance that I would like to receive assistance with.

Any help is very much appreciated!

(Question 1):
A radioactive material is decaying exponentially. The rate of change in the mass of the radioactive material is proportional to the mass. If in 3 years a mass of 60g reduces to 40g, FIND ITS HALF-LIFE.

(Question 2):
The decay of radium is proportional to its mass. If 100kg of radium takes 5 years to decay to 95kg, FIND ITS HALF-LIFE.

And finally, (Question 3):
If N=400e^(-0.4t), find t when N is HALF its initial value.

Thank you in advance to those who contribute!

#### kooliskool

##### Member
Well, first of all, this question does not give you the formula of the Q(t), so you need to derive it from knowing only the rate of change of mass is proportional to the mass. Here goes:

Let Q be the mass,

$\bg_white \frac{dQ}{dt}=kQ$
$\bg_white \int \frac{dQ}{Q}=\int kdt$
$\bg_white \ln \left | Q \right |=kt+c$

As t=0, Q=60,

$\bg_white \ln \left | 60 \right |=0k+c$
$\bg_white \therefore c=\ln 60|$

$\bg_white \therefore \ln \left | Q \right |=kt+\ln 60$

$\bg_white \therefore e^{\ln \left | Q \right |}=e^{kt+\ln 60}$
$\bg_white Q=60e^{kt}$

to find k, as t=3, Q=40,

$\bg_white \therefore 40=60e^{3k}$
$\bg_white k=\frac{1}{3}\ln \frac{2}{3}$

Now I think this is the part you got stuck I am guessing, since the question says find the half-life, it means find the time taken for the material to half itself, which means decay from 60g to 30g.

Hence, let Q=60, you will get:

$\bg_white 30=60e^{\frac{t}{3}\ln \frac{2}{3}}$
$\bg_white \ln \frac{1}{2}=\frac{t}{3}\ln \frac{2}{3}$
$\bg_white t=\frac{3\ln \frac{1}{2}}{\ln \frac{2}{3}}$

I can't be bothered to enter into the calculator, sorry.

The rest is done in similar fashion, as I have done before, try part b yourself for practice, it's exactly the same.

For c though, the tricky bit is find out it's initial value, if you don't know what it is, it's just what is N when t=0.

So:

$\bg_white N=400e^0$
$\bg_white \therefore N=400$ initially,

Hence find the time it takes to drop to 400/2=200.

Good luck

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