Question

After the release of radioactive material into the atmosphere from a nuclear power plant in a country in 1982​, the hay in that country was contaminated by a radioactive isotope​ (half-life 8 ​days). If it is safe to feed the hay to cows when 11​% of the radioactive isotope​ remains, how long did the farmers need to wait to use this​ hay?

Answer 1

Answer:

25.48 days

Step-by-step explanation:

For a radioactive substance, the Amount of the substance remaining A(t) is modeled by the equation

$A(t)=A_0(\dfrac{1}{2})^{\dfrac{t}{t_{1/2}} }$

where

$A_0 = Initial Amount \\t_{1/2}= Half-Life of the Substance \\t= Time elapsed$

In the given problem:

Half Life of the Radioactive Isotope = 8

Initial Amount, $A_0$=100%=1

A(t)=11% =0.11

Therefore substituting in the model above:

$0.11=1(\dfrac{1}{2})^{\dfrac{t}{8} }\\0.11=0.5^{t/8}\\ Change to Logarithm form \\Log_{0.5}0.11=\frac{t}{8}\\\frac{Log 0.11}{Log 0.5} =\frac{t}{8}\\3.1844=\frac{t}{8}\\t=8X3.1844=25.4752$

The farmer need to wait for 25.48 days