3. (10 points) After a large number of drinks a person has a blood alcohol level
of 200 mg/dl (milligrams per deciliter). Suppose the level of the alcohol in the
blood is an exponential function f (x) = Abx mg/dl, where x is the number of
hours elapsed. We know that after 3 hours 112.5 mg/dl remains in his blood.
Find A and b and estimate the person’s blood alcohol after 4 hours.
By what percentage is the blood alcohol level decreasing per hour?

We are given that the blood alcohol level follows an exponential function of the form:

\[
f(x) = A \cdot b^x
\]

where:
- \( f(x) \) is the blood alcohol level (mg/dl),
- \( A \) is the initial blood alcohol level,
- \( b \) is the base of the exponential function,
- \( x \) is the time elapsed (hours).

We are provided with the following information:
- At \( x = 0 \), \( f(0) = 200 \) mg/dl (initial blood alcohol level),
- After 3 hours, \( f(3) = 112.5 \) mg/dl.

### Step 1: Find \( A \)

We know that the initial blood alcohol level at \( x = 0 \) is 200 mg/dl, so:

\[
f(0) = A \cdot b^0 = A = 200
\]

Thus, \( A = 200 \).

### Step 2: Find \( b \)

Now, using the second condition that after 3 hours the blood alcohol level is 112.5 mg/dl, we can plug this into the equation:

\[
f(3) = 200 \cdot b^3 = 112.5
\]

We solve for \( b \) by dividing both sides by 200:

\[
b^3 = \frac{112.5}{200} = 0.5625
\]

Now, take the cube root of both sides to solve for \( b \):

\[
b = \sqrt[3]{0.5625} \approx 0.827
\]

Thus, \( b \approx 0.827 \).

### Step 3: Estimate blood alcohol level after 4 hours

To estimate the blood alcohol level after 4 hours, substitute \( x = 4 \), \( A = 200 \), and \( b = 0.827 \) into the equation:

\[
f(4) = 200 \cdot (0.827)^4
\]

Calculating:

\[
f(4) \approx 200 \cdot (0.467) = 93.4 \, \text{mg/dl}
\]

So, the blood alcohol level after 4 hours is approximately 93.4 mg/dl.

### Step 4: Find the percentage decrease per hour

The percentage decrease per hour can be calculated using the base \( b = 0.827 \). The rate of decrease is:

\[
\text{Percentage decrease per hour} = (1 - b) \times 100
\]

Substituting \( b = 0.827 \):

\[
\text{Percentage decrease per hour} = (1 - 0.827) \times 100 = 0.173 \times 100 = 17.3\%
\]

Thus, the blood alcohol level is decreasing by approximately 17.3% per hour.

---

Do you want more details or have any questions about this solution?

### Related Questions:
1. How would the result change if the initial blood alcohol level was higher?
2. Can you use a logarithmic function to find the rate of decay directly?
3. How would the percentage decrease per hour change if \( b \) were smaller?
4. What is the significance of the exponential model in pharmacokinetics?
5. How long would it take for the blood alcohol level to drop below 50 mg/dl?

### Tip:
In exponential decay problems, the base \( b \) represents how much of the quantity remains after each time period—keeping this interpretation in mind can help visualize the process better.

After a large number of drinks, a person has a blood alcohol level of 200 mg/dl. Suppose the level of alcohol in the blood is an exponential function f(x) = A * b^x mg/dl, where x is the number of hours elapsed. We know that after 3 hours, 112.5 mg/dl remains in the blood. Find A and b and estimate the person’s blood alcohol after 4 hours. By what percentage is the blood alcohol level decreasing per hour?

This problem explores how the blood alcohol level decreases over time, modeled by the exponential decay function f(x) = A * b^x. With an initial level of 200 mg/dl, find how it decreases to 112.5 mg/dl after 3 hours, and estimate the level after 4 hours. The solution also includes the percentage decrease per hour, calculated to be 17.3%. Ideal for students in Grades 10-12.

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