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.
after 1st hr: 200 - 200/4 = 200 - 50 = 150 mg/dL remaining,
after 2nd hr: 150 - 150/4 = 150 - 37.5 = 112.5 mg/dL remaining.
To find the person's blood alcohol level after 2 hours, we need to calculate how much alcohol remains in their blood after each hour.
Given:
Initial blood alcohol level = 200 mg/dL
Decay rate = 1/4 per hour
Let's calculate the blood alcohol level after each hour:
Hour 1:
Remaining alcohol = Initial blood alcohol level - (Decay rate * Initial blood alcohol level)
= 200 - (1/4 * 200)
= 200 - 50
= 150 mg/dL
Hour 2:
Remaining alcohol = Blood alcohol level after 1 hour - (Decay rate * Blood alcohol level after 1 hour)
= 150 - (1/4 * 150)
= 150 - 37.5
= 112.5 mg/dL
Therefore, after 2 hours, the person's blood alcohol level would be 112.5 mg/dL.
To find the person's blood alcohol level after 2 hours, we can use the exponential decay formula. Let's break down the problem step by step:
Step 1: Determine the initial blood alcohol level
The initial blood alcohol level is given as 200 mg/dL.
Step 2: Calculate the decay rate
The problem states that one fourth (1/4) of the alcohol is removed every hour. This means that the decay rate, r, is 1/4, or 0.25.
Step 3: Determine the time elapsed
The problem asks for the blood alcohol level after 2 hours. Therefore, the time elapsed, t, is 2 hours.
Step 4: Use the exponential decay formula
The exponential decay formula is given by: B = B0 * e^(-rt), where B is the blood alcohol level at time t, B0 is the initial blood alcohol level, e is the base of the natural logarithm (approximately 2.71828), r is the decay rate, and t is the time elapsed.
Plugging in the values, we have:
B = 200 * e^(-0.25 * 2)
Step 5: Calculate the blood alcohol level after 2 hours
Using a calculator,
B = 200 * e^(-0.5)
B ≈ 200 * 0.60653
B ≈ 121.31 mg/dL
Therefore, the person's blood alcohol level after 2 hours is approximately 121.31 mg/dL.