1. College algebra word problem

Thorium-228 has a half life of 1.912 years. In how many years will 75% of a sample decompose? Round answer to nearest thousandth. Show all work.

2. Re: College algebra word problem

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3. Re: College algebra word problem

All I got so far is: 1/2 n =1/4
n ln (1/2)= ln 1/4
n=2
2x 1.912=3.824

4. Re: College algebra word problem

I would let $\displaystyle A(t)$ be the amount of Thorium-228 present at time $\displaystyle t$. From the information given regarding the half-life, we may then write:

$\displaystyle A(t)=A_0\left(\frac{1}{2}\right)^{\Large \frac{t}{1.912}}$

Now, set the equal to a quarter of the initial sample:

$\displaystyle A_0\left(\frac{1}{2}\right)^{\Large \frac{t}{1.912}}=\frac{1}{4}A_0$

Divide through by $\displaystyle A_0$:

$\displaystyle \left(\frac{1}{2}\right)^{\Large \frac{t}{1.912}}= \frac{1}{4}=\left(\frac{1}{2}\right)^2$

Equate exponents:

$\displaystyle \frac{t}{1.912}=2$

$\displaystyle t=3.824\quad\checkmark$