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An element with mass 490 grams decays by 18.8% per minute. How much of the element is remaining after 8 minutes, to the nearest 1oth of a gram? Answer:

An element with mass 490490 grams decays by 18.8% 18.8 \% per minute. How much of the element is remaining after 88 minutes, to the nearest 11oth of a gram?\newlineAnswer:

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Full solution

Q. An element with mass 490490 grams decays by 18.8% 18.8 \% per minute. How much of the element is remaining after 88 minutes, to the nearest 11oth of a gram?\newlineAnswer:
  1. Understand the problem: Understand the problem.\newlineWe need to calculate the remaining mass of an element after it decays by 18.8%18.8\% per minute for 88 minutes.\newlineInitial mass of the element: 490490 grams\newlineDecay rate per minute: 18.8%18.8\%\newlineTime of decay: 88 minutes\newlineWe will use the formula for exponential decay to find the remaining mass.
  2. Calculate decay factor: Calculate the decay factor per minute.\newlineThe decay factor is the percentage of the substance that remains after each minute. Since the element decays by 18.8%18.8\%, the remaining percentage each minute is 100%−18.8%=81.2%100\% - 18.8\% = 81.2\%.\newlineTo use this in calculations, we convert the percentage to a decimal by dividing by 100100.\newlineDecay factor per minute = 81.2100=0.812\frac{81.2}{100} = 0.812
  3. Apply exponential decay formula: Apply the exponential decay formula.\newlineThe formula for the remaining mass after a certain number of minutes is:\newlineRemaining mass = Initial mass ×\times (Decay factor)number of minutes^{\text{number of minutes}}\newlineLet's plug in the values:\newlineRemaining mass after 88 minutes = 490×(0.812)8490 \times (0.812)^8
  4. Calculate remaining mass: Calculate the remaining mass after 88 minutes.\newlineNow we need to calculate 0.8120.812 raised to the power of 88 and then multiply it by 490490 grams.\newlineRemaining mass after 88 minutes = 490×(0.812)8490 \times (0.812)^8\newlineUsing a calculator, (0.812)8≈0.1693(0.812)^8 \approx 0.1693 (rounded to four decimal places)\newlineRemaining mass after 88 minutes ≈490×0.1693\approx 490 \times 0.1693\newlineRemaining mass after 88 minutes 0.8120.81200 grams
  5. Round the result: Round the result to the nearest 1010th of a gram.\newlineWe round 82.955782.9557 grams to the nearest 1010th, which gives us 83.083.0 grams.

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