Questions: A radioactive substance decays according to the following function, where y0 is the initial amount present, and y is the amount present at time t (in days). y = y0 e^(-0.0162 t) Find the half-life of this substance. Do not round any intermediate computations, and round your answer to the nearest tenth. days

Solution Steps

We are given a decay function for a radioactive substance: \[ y = y_{0} e^{-0.0162 t} \] We need to find the half-life of the substance, which is the time \( t \) when the amount \( y \) is half of the initial amount \( y_{0} \).

The half-life \( t_{1/2} \) is the time at which \( y = \frac{y_{0}}{2} \). Substitute \( y = \frac{y_{0}}{2} \) into the decay equation: \[ \frac{y_{0}}{2} = y_{0} e^{-0.0162 t_{1/2}} \]

Divide both sides by \( y_{0} \): \[ \frac{1}{2} = e^{-0.0162 t_{1/2}} \]

Take the natural logarithm of both sides to solve for \( t_{1/2} \): \[ \ln\left(\frac{1}{2}\right) = \ln\left(e^{-0.0162 t_{1/2}}\right) \] \[ \ln\left(\frac{1}{2}\right) = -0.0162 t_{1/2} \] \[ t_{1/2} = \frac{\ln\left(\frac{1}{2}\right)}{-0.0162} \]

Calculate the natural logarithm and divide: \[ t_{1/2} = \frac{\ln(0.5)}{-0.0162} \approx \frac{-0.6931}{-0.0162} \approx 42.7778 \]

Round the computed half-life to the nearest tenth: \[ t_{1/2} \approx 42.8 \]

Final Answer