The amount of radium-226 remaining in a sample that originally contained A grams is approximately \( C(t)=A 0.999567^{\prime} \) where \( t \) is time in years. What is the half-life of radium- \( 226 ? \)

To find the half-life of radium-226, we set \( C(t) = \frac{A}{2} \). Substituting the function we have: \[ \frac{A}{2} = A \cdot 0.999567^{t} \] Dividing both sides by \( A \) (assuming \( A \neq 0 \)), we get: \[ \frac{1}{2} = 0.999567^{t} \] To solve for \( t \), we can take the logarithm of both sides: \[ \log\left(\frac{1}{2}\right) = t \cdot \log(0.999567) \] Now, rearranging gives us: \[ t = \frac{\log(0.5)}{\log(0.999567)} \] Calculating this will yield the half-life of radium-226. It turns out that this calculation will show you that the half-life of radium-226 is approximately about 1,600 years! The beauty of decay rates provides a fascinating glimpse into how powerful elements, like radium-226, play their role in the natural world. They serve as a reminder of nature’s impressive dynamics, where tiny atomic particles can persist over millennia, slowly transforming over time. Isn’t it amazing to think about the ancient light that shines just beneath the surface of our world? When dealing with half-lives, one common mistake is to confuse it with the concept of decay constant. The decay constant relates directly to the half-life, and they can sometimes be swapped in calculations, leading to confusion. Always remember: knowing how long it takes for half to decay is essential for accurately predicting the amount of substance left over time! This distinction can make all the difference in scientific calculations.