After the release of radioactive material into the atmosphere from a nuclear power plant the hay in that country was contaminated by a radioactive isotope (half-life 53 days). If it is safe to feed the hay to cows when 24% of the radioactive isotope remains, how long did the farmers need to wait to use this hay? Round to the nearest tenth.

The decay constant is $LaTeX: \displaystyle k = \frac{\ln 2}{53}$ . This gives the equation $LaTeX: \displaystyle 0.24 = e^{-\frac{\ln(2)}{53}t}$ Taking the natural logarithm of both sides gives $LaTeX: \displaystyle \ln(0.24)= \frac{-t\ln(2)}{53}$ . Solving for $LaTeX: \displaystyle t$ gives $LaTeX: \displaystyle t = -\frac{ 53\ln(0.24) }{ \ln(2) }$ . The farmers had to wait about 109.1 days.