Question
This morning, Kendall drank a cup of coffee that had 95 milligrams of caffeine in it. She didn't have any more caffeine for the rest of the day. Kendall read online that the amount of caffeine in her body will decrease by approximately 13% each hour.
Write an exponential equation in the form y=a(b)x that can model the amount of caffeine, y, in Kendall's body x hours after drinking the coffee.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =
To the nearest milligram, how much caffeine will be in Kendall's body after 12 hours?
Write an exponential equation in the form y=a(b)x that can model the amount of caffeine, y, in Kendall's body x hours after drinking the coffee.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =
To the nearest milligram, how much caffeine will be in Kendall's body after 12 hours?
Answers
GPT-4o mini
To formulate the exponential equation that models the amount of caffeine in Kendall's body after drinking coffee, we can start by identifying the parameters in the general form of the exponential function:
\[ y = a(b)^x \]
Where:
- \( y \) is the amount of caffeine remaining after \( x \) hours,
- \( a \) is the initial amount of caffeine (95 mg in this case),
- \( b \) is the decay factor,
- \( x \) is the number of hours that have passed.
Given that the caffeine decreases by approximately 13% each hour, this means 87% of the caffeine remains each hour. We can express this as a decimal:
\[ b = 1 - 0.13 = 0.87 \]
Now we can substitute \( a \) and \( b \) into the equation:
- \( a = 95 \)
- \( b = 0.87 \)
Thus, our equation becomes:
\[ y = 95(0.87)^x \]
Now, to find out how much caffeine will be left in Kendall's body after 12 hours, we need to substitute \( x = 12 \) into the equation:
\[ y = 95(0.87)^{12} \]
Calculating \( (0.87)^{12} \):
\[ (0.87)^{12} \approx 0.2815 \]
Now substituting this back into the equation:
\[ y \approx 95 \times 0.2815 \approx 26.7425 \]
Rounding to the nearest milligram:
\[ y \approx 27 \]
Therefore, the amount of caffeine in Kendall's body after 12 hours is approximately **27 milligrams**.
\[ y = a(b)^x \]
Where:
- \( y \) is the amount of caffeine remaining after \( x \) hours,
- \( a \) is the initial amount of caffeine (95 mg in this case),
- \( b \) is the decay factor,
- \( x \) is the number of hours that have passed.
Given that the caffeine decreases by approximately 13% each hour, this means 87% of the caffeine remains each hour. We can express this as a decimal:
\[ b = 1 - 0.13 = 0.87 \]
Now we can substitute \( a \) and \( b \) into the equation:
- \( a = 95 \)
- \( b = 0.87 \)
Thus, our equation becomes:
\[ y = 95(0.87)^x \]
Now, to find out how much caffeine will be left in Kendall's body after 12 hours, we need to substitute \( x = 12 \) into the equation:
\[ y = 95(0.87)^{12} \]
Calculating \( (0.87)^{12} \):
\[ (0.87)^{12} \approx 0.2815 \]
Now substituting this back into the equation:
\[ y \approx 95 \times 0.2815 \approx 26.7425 \]
Rounding to the nearest milligram:
\[ y \approx 27 \]
Therefore, the amount of caffeine in Kendall's body after 12 hours is approximately **27 milligrams**.