Problem 1 - Algebra 2

Question

POLICE Police use blood alcohol content (BAC) to measure the percent concentration of alcohol in a person's bloodstream. In most states, a BAC of 0.08 percent means a person is not allowed to drive. Each hour after drinking, a person's BAC may decrease by 15\(\% .\) If a person has a BAC of 0.18 , how many hours will he need to wait until he can legally drive?

Step-by-Step Solution

Answer

The person needs to wait 6 hours until they can legally drive.

Step 1: Understand the problem

We have a person with a BAC of 0.18. To legally drive, the BAC needs to be below 0.08. Each hour, the BAC decreases by 15\(\%\). Our job is to find out how many hours it will take for the BAC to fall below 0.08.

Step 2: Set up the BAC decrease equation

The BAC decreases by 15\(\%\) each hour. This means that after each hour, the BAC is \(0.85\) times what it was the previous hour. If \(h\) is the number of hours and \(BAC\) is the current BAC, we use the equation: \(BAC_{h} = BAC_{0} \times (0.85)^{h}\). Here, \(BAC_{0} = 0.18\).

Step 3: Solve the inequality for BAC

We need to find \(h\) such that \(0.18 \times (0.85)^{h} < 0.08\). This translates into the inequality \((0.85)^{h} < \frac{0.08}{0.18}\), which simplifies to \((0.85)^{h} < 0.4444\).

Step 4: Use logarithms to find h

Take the natural logarithm of both sides to solve for \(h\): \(\ln((0.85)^{h}) < \ln(0.4444)\). Using the log rule \(\ln(a^{b}) = b \ln(a)\), we get \(h \ln(0.85) < \ln(0.4444)\).

Step 5: Calculate h

Divide both sides by \(\ln(0.85)\) (noting that \(\ln(0.85)\) is negative, so the inequality flips): \(h > \frac{\ln(0.4444)}{\ln(0.85)}\). Calculate \(\ln(0.4444) \approx -0.811\) and \(\ln(0.85) \approx -0.162\), giving \(h > 5.006\).

Step 6: Interpret the result

Since \(h\) must be a whole number and it should exceed 5.006, the person must wait for at least 6 hours for their BAC to fall below 0.08.

Key Concepts

Exponential Decay

Exponential decay is a fundamental concept that describes how a quantity decreases by a consistent percentage over equal periods of time. In the case of blood alcohol content (BAC), this principle applies as the alcohol concentration in the bloodstream drops by a fixed percentage each hour. Specifically, if a BAC is reducing by 15% every hour, we describe this using the exponential decay formula:

• Initial amount: Represents the initial value before the decay begins, symbolized by \(BAC_0\).
• Decay rate: Expressed as a decimal. For a decrease of 15%, the rate of retention is 0.85 (since the remainder is 100% - 15%).
• Equation: To find the BAC after \(h\) hours, use \(BAC_{h} = BAC_{0} \times (0.85)^{h}\).

Understanding exponential decay is crucial because it shows us that the BAC doesn't decrease linearly; rather, it gets smaller at a percentage of the previous amount, making changes significant over time.
Use this formula to estimate when a process will reduce by a desired amount, such as when it's safe to drive.

Inequality Solving

Inequality solving is a vital skill in mathematics, helping to determine boundaries within which a solution may lie. In the context of our BAC problem, we need to find out when the BAC becomes legally acceptable for driving.We express this condition through an inequality:
- Our task is to find \(h\) such that: \(0.18 \times (0.85)^{h} < 0.08\).- By rearranging and using logarithms, we more easily solve this inequality using \((0.85)^{h} < 0.4444\).In general, solving inequalities involving exponents or variables can require logarithms:
- Take the natural log of both sides: \(\ln((0.85)^{h}) < \ln(0.4444)\).- Make use of logarithmic properties like \(\ln(a^{b}) = b \ln(a)\) to isolate the variable.- Remember the rule that dividing by a negative number (e.g., \(\ln(0.85)\) here) reverses the inequality direction, critical in ensuring the right solution.
By solving inequalities this way, we learn not just when a threshold is crossed, but also how long it takes under exponential decay conditions.

Blood Alcohol Content

Blood alcohol content, or BAC, is a measure of alcohol in a person’s bloodstream, expressed as a percentage. It is used extensively in law enforcement, health, and science to assess one’s sobriety and legal capacity to perform tasks like driving. Key Details on BAC include:

• Legal Limits: Most states and countries have a BAC threshold for legal driving, typically set at 0.08%.
• Influencing Factors: BAC levels are affected by various factors such as body weight, age, the amount of alcohol consumed, and time since consumption.
• Health Implications: High BAC levels can impair cognitive and motor functions, causing a risky condition known as inebriation.

When drinking, understanding how BAC functions allows individuals to make informed decisions regarding activities like driving. Since BAC decreases over time through metabolic processing, knowing the rate of this decline (here, 15% per hour) helps someone estimate when they return to within legal limits. Ensuring these guidelines are clear is vital for safety and compliance with the law.