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In a lab experiment, the decay of a radioactive isotope is being observed. At the beginning of the first day of the experiment the mass of the substance was 1500 grams and mass was decreasing by 13% per day. Determine the mass of the radioactive sample at the beginning of the 13th day of the experiment. Round to the nearest tenth (if necessary). Answer: □ grams

In a lab experiment, the decay of a radioactive isotope is being observed. At the beginning of the first day of the experiment the mass of the substance was 15001500 grams and mass was decreasing by 13% 13 \% per day. Determine the mass of the radioactive sample at the beginning of the 1313th day of the experiment. Round to the nearest tenth (if necessary).\newlineAnswer: □\square grams

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Q. In a lab experiment, the decay of a radioactive isotope is being observed. At the beginning of the first day of the experiment the mass of the substance was 15001500 grams and mass was decreasing by 13% 13 \% per day. Determine the mass of the radioactive sample at the beginning of the 1313th day of the experiment. Round to the nearest tenth (if necessary).\newlineAnswer: □\square grams
  1. Identify initial mass and decrease: Identify the initial mass and the daily percentage decrease.\newlineThe initial mass of the radioactive substance is 15001500 grams, and it decreases by 13%13\% each day.
  2. Determine decay factor: Determine the decay factor.\newlineThe decay factor is the percentage that remains after the decay has occurred. Since the mass decreases by 13%13\% each day, the decay factor is 100%−13%=87%100\% - 13\% = 87\% or 0.870.87 in decimal form.
  3. Calculate mass after each day: Calculate the mass after each day using the decay factor.\newlineThe formula for exponential decay is P(t)=P0×(decay factor)tP(t) = P_0 \times (\text{decay factor})^{t}, where P(t)P(t) is the mass at time tt, P0P_0 is the initial mass, and tt is the number of days.
  4. Apply formula for 1313th day: Apply the formula to find the mass at the beginning of the 1313th day. P(12)=1500×(0.87)12P(12) = 1500 \times (0.87)^{12}, because we want the mass at the beginning of the 1313th day, which is after 1212 complete days.
  5. Perform calculation: Perform the calculation.\newlineP(12)=1500×(0.87)12P(12) = 1500 \times (0.87)^{12}\newlineP(12)=1500×0.1369P(12) = 1500 \times 0.1369 (rounded to four decimal places)\newlineP(12)=205.35P(12) = 205.35
  6. Round result: Round the result to the nearest tenth.\newlineThe mass of the radioactive sample at the beginning of the 13th13^{\text{th}} day is approximately 205.4205.4 grams when rounded to the nearest tenth.

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