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

25.48 days

Explanation:

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

`A(t)=A_0((1)/(2))^{(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((1)/(2))^{(t)/(8) }\\0.11=0.5^(t/8)\\$Change to Logarithm form$\\Log_(0.5)0.11=(t)/(8)\\(Log 0.11)/(Log 0.5) =(t)/(8)\\3.1844=(t)/(8)\\t=8X3.1844=25.4752`

The farmer need to wait for 25.48 days