After the release of radioactive material into the atmosphere from a nuclear power plant in a country in 1986 , the hay in that country was contaminated by a radioactive isotope (half-life 6 days). If it is safe to feed the hay to cows when 15 % of the radioactive isotope remains, how long did the farmers need to wait to use this hay?

t = 16.42 days

Explanation:

From the Law of Radioactive Decay, we know that:

N = N₀ e^ (-λt)

where,

N = Final Amount of Substance

N₀ = Initial Amount of Substance

λ = Decay Constant

t = time period

For a period of one half-life, we have:

N = 0.5*N₀

t = 6 days

Therefore,

0.5*N₀ = N₀ e^ (-λ*6 days)

-λ*6days = ln (0.5)

λ = 0.1155 day⁻¹

So, now for 15% of the element, we can re-write the formula as:

0.15*N₀ = N₀ e^ (0.1155 day⁻¹ * t)

t * 0.1155 day⁻¹ = ln (0.15)

t = 16.42 days