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)^{(x)/(h)}\implies 10=36(0.5)^{(x)/(4.9)}\implies \cfrac{10}{36}=0.5^{(x)/(4.9)} \\\\\\ \cfrac{5}{18}=0.5^{(x)/(4.9)}\implies log\left( (5)/(18) \right)=log\left( 0.5^{(x)/(4.9)} \right) \\\\\\ log\left( (5)/(18) \right)={(x)/(4.9)}\cdot log\left( 0.5 \right)\implies \cfrac{4.9\cdot log\left( (5)/(18) \right)}{log(0.5)}=x`