The half-life of radium is 1690 years. If 80 grams are present now, how much will be present in 940 years? 56.57 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 940 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 = 80 grams - Half-life of radium = 1690 years - Time elapsed = 940 years Let's calculate the amount of radium present after 940 years using the formula for radioactive decay: \[ A = A_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T}} \] 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: \[ A = 80 \times \left( \frac{1}{2} \right)^{\frac{940}{1690}} \] Now, let's calculate the amount of radium present after 940 years. Calculate the value by following steps: - step0: Calculate: \(80\left(\frac{1}{2}\right)^{\frac{940}{1690}}\) - step1: Reduce the fraction: \(80\left(\frac{1}{2}\right)^{\frac{94}{169}}\) - step2: Simplify: \(80\times \frac{1}{2^{\frac{94}{169}}}\) - step3: Multiply: \(\frac{80}{2^{\frac{94}{169}}}\) - step4: Simplify: \(\frac{80}{\sqrt[169]{2^{94}}}\) - step5: Multiply by the Conjugate: \(\frac{80\sqrt[169]{2^{75}}}{\sqrt[169]{2^{94}}\times \sqrt[169]{2^{75}}}\) - step6: Multiply the numbers: \(\frac{80\sqrt[169]{2^{75}}}{2}\) - step7: Reduce the fraction: \(40\sqrt[169]{2^{75}}\) The amount of radium present after 940 years is approximately 54.4068 grams. Rounding to the nearest thousandth, the amount of radium present after 940 years is approximately 54.407 grams.