Math Problem Statement
Solution
The problem you're dealing with revolves around the elimination of alcohol from the body, following zeroth-order elimination kinetics. This means that alcohol is eliminated at a constant rate over time, regardless of its concentration in the blood.
From the information in the image:
- The rate of alcohol elimination is given as .
- A patient's blood alcohol concentration (BAC) at is .
- The BAC is measured at regular intervals, and we are tasked with finding a formula for the BAC as a function of time and determining when the BAC reaches 0 g/liter.
Part (a):
To model the BAC over time, we can use the zeroth-order elimination equation: where:
- is the initial BAC,
- is the elimination rate,
- is the time in hours.
The equation becomes:
To find when the BAC reaches 0 g/liter, set : Solving for : So, the BAC will reach 0 after approximately 6.45 hours.
Part (b):
In this part, a driver was measured with a BAC of two hours after being pulled over. Using the equation from part (a), we need to estimate the BAC when the driver was first pulled over.
Let and . Substituting this into the formula: Solving for : Thus, the driver's BAC was approximately when they were first pulled over.
Let me know if you'd like further details or clarifications.
Here are 5 follow-up questions you might find useful:
- How does the concept of zeroth-order kinetics apply to other substances besides alcohol?
- What factors could alter the elimination rate of alcohol in the body?
- How would the equation change if the elimination followed first-order kinetics instead?
- Could external factors like body weight or food intake affect the initial BAC?
- How can the BAC equation be used in real-life situations, like estimating the time needed to be sober?
Tip: Always double-check the units used in any rate equations to ensure consistent and accurate results!
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Math Problem Analysis
Mathematical Concepts
Zeroth-order kinetics
Linear equations
Recursion
Formulas
BAC(t) = BAC_0 - k * t
t = BAC_0 / k
Theorems
Zeroth-order reaction kinetics
Suitable Grade Level
College/University Level (Introductory Chemistry or Biochemistry)
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