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After a certain medicine is ingested, its concentration in the bloodstream changes over time. The relationship between the elapsed time, t, in minutes, since the medicine was ingested, and its concentration in the bloodstream, C_("minute ")(t), in mg//L, is modeled by the following function: C_("minute ")(t)=61*(0.96)^(t) Complete the following sentence about the hourly rate of change in the medicine concentration. Round your answer to two decimal places. Every hour, the medicine concentration decays by a factor of

After a certain medicine is ingested, its concentration in the bloodstream changes over time.\newlineThe relationship between the elapsed time, t t , in minutes, since the medicine was ingested, and its concentration in the bloodstream, Cminute (t) C_{\text {minute }}(t) , in mg/L \mathrm{mg} / \mathrm{L} , is modeled by the following function:\newlineCminute (t)=61⋅(0.96)t C_{\text {minute }}(t)=61 \cdot(0.96)^{t} \newlineComplete the following sentence about the hourly rate of change in the medicine concentration. Round your answer to two decimal places.\newlineEvery hour, the medicine concentration decays by a factor of □\square

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Q. After a certain medicine is ingested, its concentration in the bloodstream changes over time.\newlineThe relationship between the elapsed time, t t , in minutes, since the medicine was ingested, and its concentration in the bloodstream, Cminute (t) C_{\text {minute }}(t) , in mg/L \mathrm{mg} / \mathrm{L} , is modeled by the following function:\newlineCminute (t)=61⋅(0.96)t C_{\text {minute }}(t)=61 \cdot(0.96)^{t} \newlineComplete the following sentence about the hourly rate of change in the medicine concentration. Round your answer to two decimal places.\newlineEvery hour, the medicine concentration decays by a factor of □\square
  1. Understand function: Understand the given function.\newlineThe function Cminute(t)=61×(0.96)tC_{\text{minute}}(t)=61\times(0.96)^{t} describes how the concentration of the medicine changes over time in minutes. The base of the exponent, 0.960.96, represents the decay factor per minute.
  2. Convert time to hours: Convert the time from minutes to hours.\newlineSince we are interested in the hourly rate of change, we need to find the decay factor for one hour. There are 6060 minutes in an hour, so we need to evaluate the decay factor for t=60t = 60.
  3. Calculate hourly decay factor: Calculate the hourly decay factor.\newlineWe substitute t=60t = 60 into the function to find the concentration after one hour.\newlineCminute(60)=61×(0.96)60C_{\text{minute}}(60) = 61 \times (0.96)^{60}
  4. Perform calculation: Perform the calculation.\newlineUsing a calculator, we raise 0.960.96 to the power of 6060.\newline(0.96)60≈0.086352314(0.96)^{60} \approx 0.086352314
  5. Find concentration after one hour: Multiply the result by the initial concentration to find the concentration after one hour.\newlineCminute(60)=61×0.086352314C_{\text{minute}}(60) = 61 \times 0.086352314\newlineCminute(60)≈5.26749118389 mg/LC_{\text{minute}}(60) \approx 5.26749118389 \, \text{mg/L}
  6. Determine decay factor: Determine the decay factor for one hour.\newlineTo find the decay factor, we need to compare the concentration after one hour to the initial concentration.\newlineDecay factor = Cminute(60)Initial concentration\frac{C_{\text{minute}}(60)}{\text{Initial concentration}}\newlineDecay factor ≈5.26749118389 mg/L61 mg/L\approx \frac{5.26749118389 \, \text{mg/L}}{61 \, \text{mg/L}}\newlineDecay factor ≈0.08635231449\approx 0.08635231449
  7. Round decay factor: Round the decay factor to two decimal places. The decay factor rounded to two decimal places is approximately 0.090.09.

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