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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%

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:

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

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

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

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

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

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

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

Hence:

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

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

c) $\displaystyle P(10)=100\cdot2^{-\frac{10}{6}}=100\cdot2^{-\frac{5}{3}}\approx31$

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.

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

It is done with $\displaystyle \LaTeX$.

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