Chapter 5.8, Problem 21AYU

To determine

To find:

The farmers need to wait this hay

Answer to Problem 21AYU

  $\frac{80}{3}=t$

Explanation of Solution

Given:

After the release of radioactive material in to the atmosphere from a nuclear power plant the hay was contaminated by iodine

  $131$ (half-life , $8days$ )

Concept used:

  $A=a.{\left(\frac{1}{2}\right)}^h$

Calculation:

Half life formula

  $\begin{array}{l}A=a.{\left(\frac{1}{2}\right)}^h\\ 10=100.{\left(\frac{1}{2}\right)}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}\\ \frac{10}{100}=\frac{100.{\left(0.5\right)}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}}{100}\\ 0.1=\frac{100.{\left(0.5\right)}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}}{100}\end{array}$

Where

A = amount left after t time

a = initial time

h = half-time

taking log on both sides

  $\begin{array}{l}0.1=\frac{100.{\left(0.5\right)}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}}{100}\\ \ln \left(0.1\right)=\ln {\left(0.5\right)}^{\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}\\ \frac{\ln \left(0.1\right)}{\ln \left(0.5\right)}\times 8=\frac{t}{8}\times 8\\ \frac{800}{30}=t\\ \frac{80}{3}=t\end{array}$