Respuesta :
Answer:
It would take 54 minutes to the element X to decay to 15 grams.
Step-by-step explanation:
A radioactive half-life refers to the amount of time it takes for half of the original isotope to decay and its given by
[tex]N(t)=N_0(\frac{1}{2})^\frac{t}{t_{1/2}}[/tex]
where,
[tex]N(t)[/tex] = quantity of the substance remaining
[tex]N_0[/tex] = initial quantity of the substance
[tex]t[/tex] = time elapsed
[tex]t_{1/2}[/tex] = half life of the substance
We know that the element X decays radioactively with a half life of 13 minutes ([tex]t_{1/2}[/tex]), there are 260 grams of it ([tex]N_0[/tex]) and we want to find how long ([tex]t[/tex]) would it take the element to decay to 15 grams ([tex]N(t)[/tex]).
Using the above formula and solving for [tex]t[/tex], we get that
[tex]15=260(\frac{1}{2})^\frac{t}{13}\\\\260\left(\frac{1}{2}\right)^{\frac{t}{13}}=15\\\\\left(\frac{1}{2}\right)^{\frac{t}{13}}=\frac{3}{52}\\\\\ln \left(\left(\frac{1}{2}\right)^{\frac{t}{13}}\right)=\ln \left(\frac{3}{52}\right)\\\\\frac{t}{13}\ln \left(\frac{1}{2}\right)=\ln \left(\frac{3}{52}\right)\\\\t=-\frac{13\ln \left(\frac{3}{52}\right)}{\ln \left(2\right)} \approx 54 \:min[/tex]
