Problem 75 - Algebra and Trigonometry with Analytic Geometry

Question

In a survey of 15 cities ranging in population \(P\) from 300 to \(3,000,000\), it was found that the average walking speed \(S\) (in \(\mathrm{ft} / \mathrm{sec})\) of a pedestrian could be approximated by \(S=0.05+0.86 \log P\). (a) How does the population affect the average walking speed? (b) For what population is the average walking speed \(5 \mathrm{ft} / \mathrm{sec}\) ?

Step-by-Step Solution

Answer

(a) As population increases, average walking speed increases. (b) Population is about 570,000 for an average walking speed of 5 ft/sec.

Step 1: Understanding the Equation

The given equation is \( S = 0.05 + 0.86 \log P \), where \( S \) is the average walking speed and \( P \) is the population. This equation shows a logarithmic relationship between population and walking speed.

Step 2: Analyzing Population Effect

Since \( S \) increases with the logarithm of \( P \), as the population \( P \) increases, the average walking speed \( S \) also increases. However, this increase is at a decreasing rate because \( \log{P} \) grows slowly for large \( P \).

Step 3: Setting Up the Equation for Part (b)

We need to find population \( P \) when the average walking speed \( S \) is \( 5 \mathrm{ft/sec} \). Set \( S = 5 \) in the equation: \[ 5 = 0.05 + 0.86 \log P \]

Step 4: Solving the Equation for \( \log P \)

Subtract 0.05 from both sides to isolate the \( \log P \) term: \[ 5 - 0.05 = 0.86 \log P \] This simplifies to \( 4.95 = 0.86 \log P \).

Step 5: Isolating \( \log P \)

Divide both sides by 0.86 to solve for \( \log P \): \[ \log P = \frac{4.95}{0.86} \approx 5.756 \]

Step 6: Solving for Population \( P \)

Since \( \log P = 5.756 \), we calculate \( P \) by finding the antilogarithm: \[ P = 10^{5.756} \approx 570000 \]

Step 7: Conclusion

Thus, for the average walking speed to be \( 5 \mathrm{ft/sec} \), the population must be approximately 570,000.

Key Concepts

Understanding the Impact of Population Growth on Walking Speed

The study highlighted in the exercise demonstrates a fascinating link between population size and average walking speed. The relationship is depicted through the logarithmic function: \[ S = 0.05 + 0.86 \log P \] where \( S \) indicates the walking speed and \( P \) represents the population.

• As the population increases, the average walking speed, \( S \), also increases.
• The growth in walking speed doesn't linearly correspond to population growth; instead, it follows a logarithmic pattern.
• This means that while larger populations see higher walking speeds, the rate of this increase diminishes as populations become very large, due to the nature of logarithms.

Understanding this logarithmic relation is vital because it helps explain how the dynamics of urban environments affect human activities like walking.

Calculating Walking Speed from Population Data

The equation \( S = 0.05 + 0.86 \log P \) offers a practical tool for predicting average walking speeds based on population size. But how does this translate into real-world calculations?

• Let's say we want to compute the expected walking speed for a specific city population.
• We substitute the population value into the formula to determine the average walking speed \( S \).
• This calculation is straightforward when given \( P \), as we're simply evaluating a mathematical expression.

The utility of this calculation lies in urban planning and public health, where knowing pedestrian dynamics can inform infrastructure and policy decisions.

Solving Logarithmic Equations in Context

Logarithmic equations can often seem daunting, but breaking them down into steps makes them more manageable. For instance, solving for population \( P \) such that the walking speed \( S \) is \( 5 \text{ ft/sec} \) involves the following steps: 1. Start with the equation: \[ 5 = 0.05 + 0.86 \log P \] 2. Isolate \( \log P \) by subtracting 0.05: \[ 4.95 = 0.86 \log P \]3. Solve for \( \log P \) by dividing by 0.86: \[ \log P = \frac{4.95}{0.86} \approx 5.756 \]4. Find \( P \) using the antilogarithm: \[ P = 10^{5.756} \approx 570000 \]

• This step-by-step approach clarifies each part of the solving process.
• Understanding the role of each component helps in solving any logistic equation you might encounter, not just this specific one.

By demystifying logarithmic equations, students and professionals can apply these skills in various analytical fields.