Question
- Scientifically, the half-life of caffeine in a human body is about 6 hours. Sophie
2 cups of coffee 10 hours ago.
How much is left in her system?
Hint: The equation of half-life is
.
A. About 0.09 of the original amount
About 0.19 of the original amount
About 0.31 of the original amount
About 0.38 of the original amount
2 cups of coffee 10 hours ago.
How much is left in her system?
Hint: The equation of half-life is
A. About 0.09 of the original amount
About 0.19 of the original amount
About 0.31 of the original amount
About 0.38 of the original amount
Ask by Reyes Vaughn. in Egypt
Jan 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
About 0.31 of the original amount is left in Sophie’s system.
Solution
To find out how much caffeine is left in Sophie’s system after 10 hours, we can use the half-life equation
, where:
-
is the amount of caffeine left in the system at time , -
is the initial amount of caffeine, -
is the decay constant (related to the half-life), -
is the time in hours.
Given that the half-life of caffeine is 6 hours, we can find the decay constant
using the formula:
Substitute the given half-life of 6 hours into the formula to find
:
Now, we can use the half-life equation to find the amount of caffeine left in Sophie’s system after 10 hours. Substitute the values of
(initial amount of caffeine),
(decay constant), and
(time) into the equation:
Since Sophie had 2 cups of coffee 10 hours ago, the initial amount of caffeine
is 2 cups. Substitute this value into the equation:
Now, we can calculate the amount of caffeine left in Sophie’s system after 10 hours.
Calculate the value by following steps:
Calculate the value by following steps:
- step0: Calculate:
- step1: Multiply the numbers:
- step2: Transform the expression:
- step3: Calculate:
- step4: Express with a positive exponent:
- step5: Simplify:
- step6: Rewrite the expression:
- step7: Multiply by the Conjugate:
- step8: Simplify:
- step9: Multiply the numbers:
- step10: Reduce the fraction:
The amount of caffeine left in Sophie’s system after 10 hours is approximately 0.63 cups, which is about 0.31 of the original amount of 2 cups of coffee.
Answered by UpStudy AI and reviewed by a Professional Tutor
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The Deep Dive
To solve the problem, we first need to understand that 10 hours is equivalent to 1.67 half-lives of caffeine (since
). Each half-life, the amount of caffeine in her system reduces by half. If Sophie started with 2 cups of coffee, we can calculate the remaining amount after 1.67 half-lives using the formula:
Remaining amount = Initial amount × (1/2)^(number of half-lives).
Thus, Sophie has about
of the original amount left.
Caffeine clearance is a super fascinating topic in pharmacology! Different factors like age, liver function, pregnancy, and even genetics can influence how long caffeine stays in one’s system. For a fun twist, some people find that regular coffee drinkers develop a tolerance to caffeine, leading to them metabolizing it faster. So, next time you sip your coffee, remember it might just be racing against the clock in your body!