# How is the percentage decay of uranium found with the equation below?

• October 27th 2012, 05:57 PM
Julianos
How is the percentage decay of uranium found with the equation below?
How would you go about solving this? (it's question 13 in the image)
The decay of uranium is modelled by D=D₀ x 2^-kt. If it takes 6 years for the mass of uranium to half, find the percentage remaining after
a) 2 years
b) 5 years
c) 10 years
The answers are a) 79%, b) 56% and c) 31%
• October 27th 2012, 06:16 PM
MarkFL
Re: How is the percentage decay of uranium found with the equation below?
We are told the half-life is 6 years, so we may state:

$\frac{1}{2}D_0=D_0\cdot2^{-k(6)}$

$\frac{1}{2}=2^{-6k}$

$\ln\left(\frac{1}{2} \right)=-6k\ln(2)$

$k=\frac{1}{6}$ and so:

$D(t)=D_0\cdot2^{-\frac{t}{6}}$

The percentage $P$ remaining after time $t$ is then:

$P(t)=\frac{D(t)}{D_0}\cdot100=\frac{100D_0\cdot2^{-\frac{t}{6}}}{D_0}=100\cdot2^{-\frac{t}{6}}$

Hence:

a) $P(2)=100\cdot2^{-\frac{2}{6}}=100\cdot2^{-\frac{1}{3}}\approx79$

b) $P(5)=100\cdot2^{-\frac{5}{6}}\approx56$

c) $P(10)=100\cdot2^{-\frac{10}{6}}=100\cdot2^{-\frac{5}{3}}\approx31$
• October 29th 2012, 12:31 AM
Julianos
Re: How is the percentage decay of uranium found with the equation below?
Also, how did you get the unique text and layout in your equations? I'd like to be able to do that myself.
• October 29th 2012, 12:38 AM
MarkFL
Re: How is the percentage decay of uranium found with the equation below?
It is done with $\LaTeX$.

Do a search here and online and you will find plenty of information and tutorials on its usage.