The half-life of radium is 1690 years. If 90 grams are present now, how much will be present in 650 years? \( \square \) grams (Do not round until the final answer. Then round to the nearest thousandth as needed.)

To find out how much radium will be present in 650 years, we can use the concept of half-life. The half-life of radium is given as 1690 years, which means that every 1690 years, the amount of radium present will be reduced by half. Given: - Initial amount of radium = 90 grams - Half-life of radium = 1690 years - Time elapsed = 650 years We can use the formula for radioactive decay to find the amount of radium present after 650 years: \[ A = A_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T}} \] Where: - \( A \) is the amount of radium present after time \( t \) - \( A_0 \) is the initial amount of radium - \( t \) is the time elapsed - \( T \) is the half-life of radium Substitute the given values into the formula: \[ A = 90 \times \left( \frac{1}{2} \right)^{\frac{650}{1690}} \] Now, we can calculate the amount of radium present after 650 years. Calculate the value by following steps: - step0: Calculate: \(90\left(\frac{1}{2}\right)^{\frac{650}{1690}}\) - step1: Reduce the fraction: \(90\left(\frac{1}{2}\right)^{\frac{5}{13}}\) - step2: Evaluate the power: \(90\times \frac{\sqrt[13]{256}}{2}\) - step3: Multiply: \(45\sqrt[13]{256}\) The amount of radium present after 650 years will be approximately 68.938486 grams. Rounding to the nearest thousandth, the amount of radium present after 650 years will be approximately 68.938 grams.