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The mass of an isotope decreases at a rate that is proportional to the mass at that time. The mass of the isotope was 40 grams initially, and it was 10 grams after 16 days. What was the mass of the isotope after 20 days? Round to the nearest gram. grams

The mass of an isotope decreases at a rate that is proportional to the mass at that time.\newlineThe mass of the isotope was 4040 grams initially, and it was 1010 grams after 1616 days.\newlineWhat was the mass of the isotope after 2020 days?\newlineRound to the nearest gram.\newline□\square grams

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Q. The mass of an isotope decreases at a rate that is proportional to the mass at that time.\newlineThe mass of the isotope was 4040 grams initially, and it was 1010 grams after 1616 days.\newlineWhat was the mass of the isotope after 2020 days?\newlineRound to the nearest gram.\newline□\square grams
  1. Identify Mass and Time: Identify the initial mass, the mass after a certain time, and the time elapsed.\newlineInitial mass m0m_0 = 4040 grams\newlineMass after 1616 days m16m_{16} = 1010 grams\newlineTime elapsed for m16m_{16} = 1616 days\newlineWe need to find the mass after 2020 days m20m_{20}.
  2. Use Exponential Decay Formula: Use the formula for exponential decay, which is m(t)=m0⋅e−ktm(t) = m_0 \cdot e^{-kt}, where m(t)m(t) is the mass at time tt, m0m_0 is the initial mass, kk is the decay constant, and tt is the time.\newlineWe need to find the value of kk using the information that m(16)=10m(16) = 10 grams.
  3. Substitute Values and Solve: Substitute the known values into the exponential decay formula to find kk.10=40⋅e−16k10 = 40 \cdot e^{-16k}Divide both sides by 4040 to isolate e−16ke^{-16k}.1040=e−16k\frac{10}{40} = e^{-16k}14=e−16k\frac{1}{4} = e^{-16k}
  4. Take Natural Logarithm: Take the natural logarithm of both sides to solve for kk.ln⁡(14)=ln⁡(e−16k)\ln(\frac{1}{4}) = \ln(e^{-16k})ln⁡(14)=−16k×ln⁡(e)\ln(\frac{1}{4}) = -16k \times \ln(e)Since ln⁡(e)=1\ln(e) = 1, we have:ln⁡(14)=−16k\ln(\frac{1}{4}) = -16k
  5. Solve for k: Solve for k.\newlinek=−ln⁡(14)/16k = -\ln(\frac{1}{4}) / 16\newlinek≈0.08664k \approx 0.08664 (using a calculator)
  6. Find Mass After 2020 Days: Now that we have kk, we can find the mass after 2020 days using the same exponential decay formula.m(20)=40⋅e(−0.08664⋅20)m(20) = 40 \cdot e^{(-0.08664 \cdot 20)}Calculate the value using a calculator.m(20)≈40⋅e(−1.7328)m(20) \approx 40 \cdot e^{(-1.7328)}m(20)≈40⋅0.177m(20) \approx 40 \cdot 0.177m(20)≈7.08m(20) \approx 7.08
  7. Round to Nearest Gram: Round the result to the nearest gram. m(20)≈7m(20) \approx 7 grams

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