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Question Watch Video Show Examples An element with mass 780 grams decays by 16.3% per minute. How much of the element is remaining after 16 minutes, to the nearest 10 th of a gram?

Question\newlineWatch Video\newlineShow Examples\newlineAn element with mass 780780 grams decays by 16.3% 16.3 \% per minute. How much of the element is remaining after 1616 minutes, to the nearest 1010 th of a gram?

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Q. Question\newlineWatch Video\newlineShow Examples\newlineAn element with mass 780780 grams decays by 16.3% 16.3 \% per minute. How much of the element is remaining after 1616 minutes, to the nearest 1010 th of a gram?
  1. Identify values: Identify the initial amount, decay rate per minute, and total time.\newlineInitial amount aa = 780780 grams\newlineDecay rate per minute rr = 16.3%16.3\%\newlineTotal time tt = 1616 minutes
  2. Convert to decimal: Convert the decay rate from a percentage to a decimal.\newlineTo convert a percentage to a decimal, divide by 100100.\newliner=16.3%100=0.163r = \frac{16.3\%}{100} = 0.163
  3. Determine decay factor: Determine the decay factor per minute. The decay factor is the amount by which the substance decreases each minute. Since it decays by 16.3%16.3\%, it retains 100%−16.3%=83.7%100\% - 16.3\% = 83.7\% of its mass each minute. Decay factor per minute = 1−r=1−0.163=0.8371 - r = 1 - 0.163 = 0.837
  4. Calculate remaining amount: Calculate the remaining amount of the element after 1616 minutes.\newlineThe formula for exponential decay is:\newlineremaining amount == initial amount ×\times (decay factor)total time^{\text{total time}}\newlineSubstitute the known values into the formula:\newlineremaining amount == 780780 ×\times (0.837)16(0.837)^{16}
  5. Perform calculation: Perform the calculation.\newlineUse a calculator to compute the remaining amount:\newlineremaining amount = 780×(0.837)16780 \times (0.837)^{16}\newlineremaining amount ≈780×0.0494\approx 780 \times 0.0494 (rounded to 44 decimal places)\newlineremaining amount ≈38.532\approx 38.532 (rounded to 33 decimal places)
  6. Round to nearest tenth: Round the result to the nearest tenth of a gram.\newlineTo round to the nearest tenth, look at the hundredths place. If it is 55 or more, round up; if it is less than 55, round down.\newlineremaining amount ≈38.5\approx 38.5 grams (since the hundredths place is 33, we round down)

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