Cesium- 137 is a radioactive isotope with a variety of industrial and medical applications, known for its relatively long half-life of about 30 years. It has been observed that the activity of the sample decreases by \( 2.3 \% \) each year. This decay can be represented by the recursive equation \( B(x) \approx B(x-1) \cdot 0.977 \), in which \( B(x) \) is the activity in millicuries after \( x \) years, and \( B(0)=15 \mathrm{mCi} \). Create an exponential equation that models the decay of cesium- -137 in the sample over time. Round your answer to four decimal places. Which of the following is the correct equation? (1 point) \( B=15 e^{-0.0221 t} \) \( B=15 e^{-0.0230 t} \) \( B=15 e^{-0.0233 t} \) \( B=15(0.9770)^{t} \)

To convert the recursive equation into an exponential equation, we start with the given decay of \( B(x) \approx B(x-1) \cdot 0.977 \). This indicates that the activity decreases by a factor of \( 0.977 \) each year, which can also be expressed in the form of an exponential decay model. Using the formula for exponential decay \( B(t) = B_0 e^{-kt} \), where \( k \) is the decay constant, we can match this with the recursive form. The decay factor of \( 0.977 \) indicates that \( k \) can be calculated as follows: \[ e^{-k} = 0.977 \implies -k = \ln(0.977) \implies k \approx 0.0230. \] Thus, the equation for the activity after \( t \) years will be \( B(t) = 15 e^{-0.0230 t} \). So, the correct exponential equation that models the decay of cesium-137 is: **\( B = 15 e^{-0.0230 t} \)**. For your multiple-choice question, this corresponds to the second presented option!