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.
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.
Answers
GPT-4o mini
To model the amount of caffeine, \( y \), in Kendall's body \( x \) hours after drinking the coffee, we can use the exponential decay formula in the form \( y = a(b)^x \).
1. **Identify the initial amount of caffeine:**
\( a = 95 \) (the initial amount of caffeine in milligrams)
2. **Determine the decay rate:**
If the caffeine decreases by approximately 13% each hour, that means after each hour, 87% (or 0.87) of the caffeine remains in her body.
Therefore, \( b = 0.87 \).
3. **Construct the equation:**
The equation that models the amount of caffeine in Kendall's body after \( x \) hours is:
\[
y = 95(0.87)^x
\]
So, the final exponential equation is:
\[
y = 95(0.87)^x
\]
1. **Identify the initial amount of caffeine:**
\( a = 95 \) (the initial amount of caffeine in milligrams)
2. **Determine the decay rate:**
If the caffeine decreases by approximately 13% each hour, that means after each hour, 87% (or 0.87) of the caffeine remains in her body.
Therefore, \( b = 0.87 \).
3. **Construct the equation:**
The equation that models the amount of caffeine in Kendall's body after \( x \) hours is:
\[
y = 95(0.87)^x
\]
So, the final exponential equation is:
\[
y = 95(0.87)^x
\]