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The chemical element einsteinium253 naturally loses its mass over time. When a sample of einsteinium-253 was initially measured, it had a mass of 15 grams. The relationship between the elapsed time t, in weeks, and the mass, M_("week ")(t), left in the sample is modeled by the following function: M_("week ")(t)=15*(0.79)^(t) Complete the following sentence about the daily rate of change in the mass of the sample. Round your answer to two decimal places. Every day, the mass of the sample decays by a factor of

The chemical element einsteinium253253 naturally loses its mass over time. When a sample of einsteinium−253-253 was initially measured, it had a mass of 1515 grams.\newlineThe relationship between the elapsed time t t , in weeks, and the mass, Mweek (t) M_{\text {week }}(t) , left in the sample is modeled by the following function:\newlineMweek (t)=15⋅(0.79)t M_{\text {week }}(t)=15 \cdot(0.79)^{t} \newlineComplete the following sentence about the daily rate of change in the mass of the sample.\newlineRound your answer to two decimal places.\newlineEvery day, the mass of the sample decays by a factor of

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Q. The chemical element einsteinium253253 naturally loses its mass over time. When a sample of einsteinium−253-253 was initially measured, it had a mass of 1515 grams.\newlineThe relationship between the elapsed time t t , in weeks, and the mass, Mweek (t) M_{\text {week }}(t) , left in the sample is modeled by the following function:\newlineMweek (t)=15⋅(0.79)t M_{\text {week }}(t)=15 \cdot(0.79)^{t} \newlineComplete the following sentence about the daily rate of change in the mass of the sample.\newlineRound your answer to two decimal places.\newlineEvery day, the mass of the sample decays by a factor of
  1. Understand Function: Understand the given function and what it represents.\newlineThe function Mweek(t)=15×(0.79)tM_{\text{week}}(t)=15\times(0.79)^{t} shows the mass of einsteinium−253-253 after tt weeks. To find the daily rate of change, we need to convert the time from weeks to days.
  2. Convert Time: Convert the time from weeks to days.\newlineThere are 77 days in a week. Therefore, to express tt in days, we need to divide the number of days by 77 to get the equivalent number of weeks.\newlineLet dd be the number of days. Then tt (in weeks) is d/7d/7.
  3. Rewrite Function: Rewrite the function in terms of days. Mday(d)=15×(0.79)d7M_{\text{day}}(d) = 15 \times (0.79)^{\frac{d}{7}}
  4. Calculate Rate of Change: Calculate the daily rate of change.\newlineTo find the daily rate of change, we need to find the factor by which the mass changes from one day to the next. This is the value of the function when d=1d = 1 (one day) divided by the value of the function when d=0d = 0 (initial mass).
  5. Calculate Mass After One Day: Calculate the mass after one day. Mday (1)=15×(0.79)17M_{\text{day }}(1) = 15\times(0.79)^{\frac{1}{7}}
  6. Calculate Initial Mass: Calculate the mass after zero days (initial mass). \newlineMday (0)=15×(0.79)07=15×1=15M_{\text{day }}(0) = 15\times(0.79)^{\frac{0}{7}} = 15\times1 = 15
  7. Calculate Decay Factor: Calculate the factor of decay after one day.\newlineFactor of decay = Mday (1)Mday (0)\frac{M_{\text{day }}(1)}{M_{\text{day }}(0)}
  8. Perform Calculation: Perform the calculation.\newlineFactor of decay = (15×(0.79)17)/15(15\times(0.79)^{\frac{1}{7}}) / 15\newlineFactor of decay = (0.79)17(0.79)^{\frac{1}{7}}
  9. Use Calculator: Use a calculator to find the value of (0.79)1/7(0.79)^{1/7} and round to two decimal places.\newlineFactor of decay ≈(0.79)1/7≈0.97\approx (0.79)^{1/7} \approx 0.97 (rounded to two decimal places)

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