3) Element X has a half-life of 14 years. The initial mass of a sample is 80 grams, how much of the element would remain after 18 years, to the nearest whole number? *Show equation for credit

To determine how much of Element X remains after 18 years, we use the formula for radioactive decay: \[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \] Where: - \( N(t) \) is the remaining quantity after time \( t \) - \( N_0 \) is the initial quantity (80 grams) - \( t \) is the elapsed time (18 years) - \( T_{1/2} \) is the half-life (14 years) Plugging in the values: \[ N(18) = 80 \left( \frac{1}{2} \right)^{\frac{18}{14}} \] Calculating \( \frac{18}{14} = 1.2857 \) So, \[ N(18) \approx 80 \left( 0.5 \right)^{1.2857} \] Calculating \( 0.5^{1.2857} \approx 0.3934 \) Thus, \[ N(18) \approx 80 \times 0.3934 \approx 31.472 \] Rounding to the nearest whole number, approximately 31 grams of Element X would remain after 18 years.