After several drinks, a person has a blood alcohol level of 200 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 2 hours.
Well, well, well! Sounds like someone had a wild night, huh? Let's do some math and find out just how sober they are now.
If a person's blood alcohol level decays exponentially, with one fourth being removed every hour, that means after one hour, three-fourths (3/4) of the alcohol will remain.
So, after 1 hour, the blood alcohol level would be 200 mg/dL * (3/4) = 150 mg/dL.
Now, we need to do some more decay!
After another hour (2 hours in total), three-fourths (3/4) of the remaining alcohol will be removed. We can calculate this as:
150 mg/dL * (3/4) = 112.5 mg/dL.
And voila! After 2 hours, our party animal's blood alcohol level is approximately 112.5 mg/dL.
Remember, though, I'm just a clown bot - always make sure you drink responsibly and stay safe!
To find the person's blood alcohol level after 2 hours, we can use the formula for exponential decay:
C(t) = C(0) * e^(-kt)
Where:
C(t) is the blood alcohol level after time t
C(0) is the initial blood alcohol level (200 mg/dL)
k is the decay constant
t is the time in hours
Since one fourth of the alcohol is removed every hour, the decay constant, k, is equal to ln(1/4), which is approximately -1.3863.
Let's substitute these values into the formula and calculate the blood alcohol level after 2 hours:
C(2) = 200 * e^(-1.3863 * 2)
Using a calculator, the calculation becomes:
C(2) ≈ 200 * e^(-2.7726)
C(2) ≈ 200 * 0.0625
C(2) ≈ 12.5 mg/dL
Therefore, the person's blood alcohol level after 2 hours would be approximately 12.5 mg/dL.
To find the person's blood alcohol level after 2 hours, we need to apply the concept of exponential decay. Here's how you can calculate it:
Step 1: Determine the initial amount of alcohol in the blood.
Given that the person has a blood alcohol level of 200 mg/dL, this is our initial value.
Step 2: Determine the decay rate.
Since one fourth is removed every hour, the decay rate is 1/4 or 0.25.
Step 3: Apply the exponential decay formula.
The formula for exponential decay is A = A₀ * e^(-kt), where:
- A is the final amount after time t
- A₀ is the initial amount
- k is the decay rate
- t is the time in hours
Therefore, the formula for our problem becomes A = 200 * e^(-0.25 * 2).
Step 4: Calculate the blood alcohol level after 2 hours.
A = 200 * e^(-0.25 * 2)
= 200 * e^(-0.5)
≈ 200 * 0.6065
≈ 121.3 mg/dL
Therefore, the person's blood alcohol level after 2 hours is approximately 121.3 mg/dL.