After several drinks, a person has a blood alcohol level of 220 mg/dL (milligrams per deciliter). If the amount of alcohol in the blood decays exponentially, with one fourth being removed every hour, find the person's blood alcohol level after 4 hours
Work with what's left, i.e. r=3/4.
Over 4 hours, fraction left
x = (3/4)^4=0.3164
Level = 220 mg/dL * x = 69.6 mg/dL.
To find the person's blood alcohol level after 4 hours, we need to understand how the alcohol content in the blood decays exponentially over time.
We are given that the initial blood alcohol level is 220 mg/dL and that one fourth (25%) of the alcohol is removed every hour. This means that after each hour, the alcohol content remaining in the blood is reduced to 75% of its previous value.
To calculate the blood alcohol level after 4 hours, we can use the formula for exponential decay:
C = C₀ * (1 - r)^t
Where:
C is the blood alcohol level at time t
C₀ is the initial blood alcohol level
r is the decay rate (in this case, 25% or 0.25)
t is the time in hours
Let's plug in the values:
C = 220 * (1 - 0.25)^4
Calculating the value inside the parentheses:
(1 - 0.25) = 0.75
Raise this to the power of 4:
0.75^4 = 0.31640625
Now multiply this value by the initial blood alcohol level:
C = 220 * 0.31640625
Calculating this product:
C ≈ 69.41 mg/dL
Therefore, after 4 hours, the person's blood alcohol level would be approximately 69.41 mg/dL.