Answer to Question #209466 in Calculus for Rica

Question

The half-life of a radioactive substance is defined to be the amount of time it takes for the substance to decay 50% of its amount. If substance X has a half-life of 3,600 years, what part of substance X will remain after 4,500 years?

Answer 1

We know that ,

$A(t)=A_0 (\frac{1}{2})^{\frac{t}{T}}$

Where ,

$A_0=$ the amount initially present .

$T=$ the half life of the substance .

$t=$ the time period over which the substance is studied .

$A(t)=$ the amount of the substance present after time t .

In our case we have ,

$A_0=X, T=3600 \ and \ t=4500$

We have to calculate $A(t)=?$

Therefore according to question ,

$A(t)=X ×(\frac{1}{2})^{\frac{4500}{3600}}$

$\implies A(t)=X×(\frac{1}{2})^{\frac{5}{4}}$

$\implies A(t)=X×(0.5)^\frac{5}{4}$

$\implies A(t)\approx X×(0.420448208)$

Hence after 4500 years approximately 0.420448208 parts of the substance remain .