Question

Element X decays radioactively with a half life of 8 minutes. If there are 450 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 37 grams?

Answer 1

$t_{1/2}=8\text{ min}\\N_0=450\text{ g}\\N(t)=37\text{ g}\\t=?\\\\N(t)=N_0\left(\dfrac{1}{2}\right)^{\tfrac{t}{t_{1/2}}}\\\\37=450\cdot\left(\dfrac{1}{2}\right)^{\tfrac{t}{8}}\\\\\left(\dfrac{1}{2}\right)^{\tfrac{t}{8}}=\dfrac{37}{450}\\\\\dfrac{t}{8}=\log_{\tfrac{1}{2}}\dfrac{37}{450}\\\\t=8\log_{\tfrac{1}{2}}\dfrac{37}{450}\\\vdots\\t\approx28.8\text{ min}$