Question

The mass of a radioactive element at time x is given by the equation below, where c is the original mass and h is the half-life of the element. The half-life of a substance is 4.9 years. How long will it take for 36 grams to decay to 10 grams? (Round your answer to one decimal place.)
M(x)=c(.5^x/h)

Answer 1

$\bf M(x)=c(0.5)^{\frac{x}{h}}\implies 10=36(0.5)^{\frac{x}{4.9}}\implies \cfrac{10}{36}=0.5^{\frac{x}{4.9}} \\\\\\ \cfrac{5}{18}=0.5^{\frac{x}{4.9}}\implies log\left( \frac{5}{18} \right)=log\left( 0.5^{\frac{x}{4.9}} \right) \\\\\\ log\left( \frac{5}{18} \right)={\frac{x}{4.9}}\cdot log\left( 0.5 \right)\implies \cfrac{4.9\cdot log\left( \frac{5}{18} \right)}{log(0.5)}=x$