Problem 51 - Algebra 2

Question

CHALLENGE If one bulb in a string of holiday lights fails to work, the whole string will not light. If each bulb in a set has a 99.5\(\%\) chance of working, what is the maximum number of lights that can be strung together with at least a 90\(\%\) chance of the whole string lighting?

Step-by-Step Solution

Answer

The maximum number of bulbs is 21.

Step 1: Understanding the problem

The challenge is to find the maximum number of bulbs in a string such that the probability of all bulbs working is at least 90\(\%\). Each bulb has a 99.5\(\%\) probability of functioning properly.

Step 2: Define the probability equation

For a string of \( n \) bulbs to all work, the probability is given by \( (0.995)^n \). We want \( (0.995)^n \geq 0.90 \).

Step 3: Solve for \( n \)

To find \( n \), we take the logarithm of both sides: \( \ln((0.995)^n) \geq \ln(0.90) \). This simplifies to \( n \cdot \ln(0.995) \geq \ln(0.90) \).

Step 4: Isolate \( n \)

Solve the inequality for \( n \): \( n \leq \frac{\ln(0.90)}{\ln(0.995)} \).

Step 5: Calculate \( n \)

Calculate \( n \) using a calculator: \[ n \leq \frac{\ln(0.90)}{\ln(0.995)} \approx \frac{-0.10536}{-0.00501} \approx 21.055. \] Since \( n \) must be an integer, the maximum \( n \) is 21.

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics, important for solving problems involving exponential relationships, such as the exercise on holiday lights. When dealing with probabilities that involve a power, like \((0.995)^n\), logarithms become a useful tool for simplifying the calculation.

A logarithm answers the question: how many times do we need to multiply a certain base to get another number? For instance, \(\log_b(x)\) gives us the power to which the base \(b\) must be raised to obtain \(x\).

• The natural logarithm, denoted by \(\ln(x)\), is a common logarithm with the base \(e \approx 2.718\). It is particularly useful in continuous growth and decay models.


• To solve equations like \((0.995)^n \geq 0.90\), we use logarithms to "bring down" the exponent \(n\), allowing us to manipulate and solve the inequality algebraically.


• Using \(\ln((0.995)^n) = n \cdot \ln(0.995)\) clarifies our task, enabling us to solve for \(n\) efficiently.

Understanding how logarithms function, and applying them, empowers us to solve complex problems not easily tackled by basic algebraic methods.

Inequality Solving

Solving inequalities is a core mathematical skill that's crucial when we need to find feasible solutions within certain bounds. In our holiday lights problem, we set up the inequality \( (0.995)^n \geq 0.90 \) to find the maximum number of bulbs that maintain a 90\(\%\) operational probability.

Here's a simple approach to solving such inequalities:

• Start by rewriting the problem using logarithms, which turns the multiplicative relationship into an additive one: \(\ln((0.995)^n) \geq \ln(0.90)\).


• Next, apply properties of logarithms, such as \(\ln(a^b) = b \cdot \ln(a)\), to simplify the inequality: \(n \cdot \ln(0.995) \geq \ln(0.90)\).


• Isolate \(n\) by dividing both sides of the inequality by \(\ln(0.995)\), thus finding \(n \leq \frac{\ln(0.90)}{\ln(0.995)}\).

The trick is in handling the inequality throughout the steps, ensuring the mathematical properties are respected. Since \(n\) is quantified as below the continuous value calculated, rounding down provides the solution: 21 in this context.

Exponential Functions

Exponential functions describe processes that grow or decay at a constant relative rate, a concept frequently encountered in probability, growth models, and more. In the holiday lights problem, the function \((0.995)^n\) represents the probability that all \(n\) bulbs work. Each bulb works independently, with a high probability of functioning, forming a pattern of decay over a series of probabilities.

These characteristics of exponential functions make them suitable for modeling real-world situations like chain reliability of bulbs. Key features of exponential functions include:

• Rapid growth or decay: Even small changes in the exponent \(n\) can significantly alter the outcome. This highlights sensitivity which is seen as the number of bulbs increases, the probability of function dropping sharply.


• The base of the exponential function (in this case, 0.995): This represents the rate at which changes occur, indicating each bulb's reliability affects the entire chain.

By comprehending these functions, we efficiently predict how changing variables influence outcomes in compound systems, highlighting the exponential decay pattern evident as more bulbs are added to our string.