Element X is a radioactive isotope such that every 16 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 5900 grams, how much of the element would remain after 27 years,to the nearest whole number?

## Step 1: Determine the number of half-lives<br />### Divide the total time elapsed (27 years) by the half-life duration (16 years) to calculate how many half-lives have passed:<br />\[<br />\text{Number of half-lives} = \frac{27}{16} = 1.6875<br />\]<br />This means 1 full half-life has passed, and a fraction of another half-life remains.<br /><br />## Step 2: Apply the decay formula<br />### Use the radioactive decay formula:<br />\[<br />M = M_0 \cdot \left(\frac{1}{2}\right)^n<br />\]<br />where:<br />- \(M_0 = 5900\) grams (initial mass),<br />- \(n = 1.6875\) (number of half-lives).<br /><br />Substitute the values:<br />\[<br />M = 5900 \cdot \left(\frac{1}{2}\right)^{1.6875}<br />\]<br /><br />## Step 3: Simplify the calculation<br />### Compute the value of \(\left(\frac{1}{2}\right)^{1.6875}\):<br />\[<br />\left(\frac{1}{2}\right)^{1.6875} \approx 0.3118<br />\]<br /><br />Now multiply:<br />\[<br />M = 5900 \cdot 0.3118 \approx 1839.62<br />\]<br /><br />## Step 4: Round to the nearest whole number<br />### The remaining mass is approximately \(1840\) grams after rounding.