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Element X is a radioactive isotope such that every 75 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 920 grams, how much of the element would remain after 3 years, to the nearest whole number? Answer:

Element X \mathrm{X} is a radioactive isotope such that every 7575 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 920920 grams, how much of the element would remain after 33 years, to the nearest whole number?\newlineAnswer:

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Q. Element X \mathrm{X} is a radioactive isotope such that every 7575 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 920920 grams, how much of the element would remain after 33 years, to the nearest whole number?\newlineAnswer:
  1. Determine half-life of Element X: Determine the half-life of Element X. The half-life of Element X is given as 7575 years. This means that every 7575 years, the mass of Element X is reduced by half.
  2. Calculate number of half-lives: Calculate the number of half-lives that occur in 33 years.\newlineSince the half-life is 7575 years, we need to find out how many 7575-year periods fit into 33 years. This is done by dividing 33 by 7575.\newlineNumber of half-lives = 33 years / 7575 years = 0.040.04 half-lives (approximately).
  3. Use exponential decay formula: Use the exponential decay formula to find the remaining mass.\newlineThe formula for exponential decay is:\newlineFinal mass = Initial mass ×\times (1/2)Number of half-lives(1/2)^{\text{Number of half-lives}}\newlineWe will use the number of half-lives calculated in Step 22 to find the remaining mass after 33 years.
  4. Plug values into formula: Plug the values into the exponential decay formula.\newlineFinal mass = 920920 grams ×(12)0.04\times \left(\frac{1}{2}\right)^{0.04}\newlineTo calculate this, we need to evaluate (12)0.04\left(\frac{1}{2}\right)^{0.04}.
  5. Calculate (12)0.04(\frac{1}{2})^{0.04}: Calculate (12)0.04(\frac{1}{2})^{0.04}. Using a calculator, we find that (12)0.04(\frac{1}{2})^{0.04} is approximately 0.9710.971.
  6. Multiply initial mass: Multiply the initial mass by the decay factor to find the final mass.\newlineFinal mass = 920920 grams ×0.971\times 0.971\newlineFinal mass ≈893.32\approx 893.32 grams
  7. Round final mass: Round the final mass to the nearest whole number.\newlineThe final mass of Element XX after 33 years, rounded to the nearest whole number, is approximately 893893 grams.

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