Problem 105
Question
Bornstein and Bornstein found in a study that the average walking speed \(v,\) in feet per second, of a person living in a city of population \(p,\) in thousands, is $$v(p)=0.37 \ln p+0.05 $$. a) The population of Seattle is \(635,000(p=635)\). What is the average walking speed of a person living in Seattle? b) The population of New York is \(8,340,000 .\) What is the average walking speed of a person living in New York? c) Find \(v^{\prime}(p)\). d) Interpret \(v^{\prime}(p)\) found in part (c).
Step-by-Step Solution
Verified Answer
a) 2.438 feet/sec b) 3.390 feet/sec c) \( \frac{0.37}{p} \) d) Walking speed increases at a decreasing rate with population.
1Step 1: Convert Population to Thousands
First, convert the population of Seattle from individuals to thousands. Given: \[ p = 635,000 \]Since the formula uses populations in thousands, we have:\[ p = 635 \]
2Step 2: Calculate Walking Speed for Seattle
Substitute \( p = 635 \) into the formula for average walking speed:\[ v(635) = 0.37 \ln(635) + 0.05 \]Calculate the natural logarithm:\[ \ln(635) \approx 6.454 \]So,\[ v(635) = 0.37 \times 6.454 + 0.05 = 2.388 + 0.05 = 2.438 \text{ feet per second} \]
3Step 3: Convert Population for New York to Thousands
Convert the population of New York from individuals to thousands:\[ p = 8,340,000 \]Thus, the population in thousands is:\[ p = 8340 \]
4Step 4: Calculate Walking Speed for New York
Substitute \( p = 8340 \) into the formula for average walking speed:\[ v(8340) = 0.37 \ln(8340) + 0.05 \]Calculate the natural logarithm:\[ \ln(8340) \approx 9.028 \]So,\[ v(8340) = 0.37 \times 9.028 + 0.05 = 3.340 + 0.05 = 3.390 \text{ feet per second} \]
5Step 5: Differentiate the Walking Speed Function
To find \( v'(p) \), differentiate \( v(p) = 0.37 \ln p + 0.05 \) with respect to \( p \):\[ v'(p) = \frac{d}{dp}[0.37 \ln p + 0.05] \]The derivative of \( \ln p \) is \( \frac{1}{p} \), therefore:\[ v'(p) = 0.37 \times \frac{1}{p} = \frac{0.37}{p} \]
6Step 6: Interpret the Derivative
The derivative \( v'(p) = \frac{0.37}{p} \) represents the rate of change of walking speed with respect to the population of a city. It means that as the population increases, the walking speed increases at a rate inversely proportional to the population size, meaning larger populations increase walking speed at a decreasing rate.
Key Concepts
Population ModelingDerivativesNatural Logarithms
Population Modeling
In calculus and urban studies, population modeling is essential to understand how various factors impact behaviors within a city. Population modeling involves creating equations or models that describe how certain variables, such as average walking speed, vary with the population size. Consider the function given in the exercise:
\[v(p) = 0.37 \ln p + 0.05\]This equation models the average walking speed \(v\) of a person living in a city as a function of the city's population \(p\), measured in thousands. The term \(\ln p\) indicates the influence of the city's population on walking speed, while the constants 0.37 and 0.05 fine-tune the model to align with observed data.
By structuring the model in this way, we capture the intuitive idea that larger city populations may result in faster walking speeds, possibly due to greater efficiency in pedestrian movement necessary in crowded environments. This simple linear growth aids urban planners and researchers by predictively analyzing how changes in population could affect city dynamics.
\[v(p) = 0.37 \ln p + 0.05\]This equation models the average walking speed \(v\) of a person living in a city as a function of the city's population \(p\), measured in thousands. The term \(\ln p\) indicates the influence of the city's population on walking speed, while the constants 0.37 and 0.05 fine-tune the model to align with observed data.
By structuring the model in this way, we capture the intuitive idea that larger city populations may result in faster walking speeds, possibly due to greater efficiency in pedestrian movement necessary in crowded environments. This simple linear growth aids urban planners and researchers by predictively analyzing how changes in population could affect city dynamics.
Derivatives
In calculus, derivatives are crucial for understanding how a function's output changes with respect to changes in its input. In this context, when examining walking speed related to population size, taking the derivative helps us quantify the sensitivity of walking speed to population changes.
**Understanding Derivatives**- The derivative, denoted as \(v'(p)\), gives the rate at which the walking speed changes as the population increases. - For the function \(v(p) = 0.37 \ln p + 0.05\), computing the derivative \(v'(p)\) involves applying the chain rule and recognizing that the derivative of \(\ln p\) with respect to \(p\) is \(\frac{1}{p}\).
The calculated derivative is thus:\[v'(p) = \frac{0.37}{p}\]This derivative implies that as the population \(p\) increases, each additional person contributes less to the increase in walking speed than the one before, reflecting a diminishing return effect. Such insights demonstrate how derivatives provide deeper understanding of the modeled phenomena beyond just computing values.
**Understanding Derivatives**- The derivative, denoted as \(v'(p)\), gives the rate at which the walking speed changes as the population increases. - For the function \(v(p) = 0.37 \ln p + 0.05\), computing the derivative \(v'(p)\) involves applying the chain rule and recognizing that the derivative of \(\ln p\) with respect to \(p\) is \(\frac{1}{p}\).
The calculated derivative is thus:\[v'(p) = \frac{0.37}{p}\]This derivative implies that as the population \(p\) increases, each additional person contributes less to the increase in walking speed than the one before, reflecting a diminishing return effect. Such insights demonstrate how derivatives provide deeper understanding of the modeled phenomena beyond just computing values.
Natural Logarithms
Natural logarithms (ln) are logarithms with the base of Euler's number \(e\), approximately 2.718. They frequently appear in natural and social sciences due to their logarithmic scale properties, which enable transforming multiplicative processes into additive ones, simplifying complex phenomena.
**Significance of Natural Logarithms in Modeling**- In our exercise, the natural logarithm \(\ln p\) is used, helping to model the average walking speed's natural growth relative to urban population size.- The model’s form \(v(p) = 0.37 \ln p + 0.05\) uses \(\ln p\) to express how initially, for smaller populations, changes in population size have a more substantial effect on walking speed. As a result, it captures the decreasing return on the walking speed increase as the population grows.
Utilizing natural logarithms provides a more realistic portrayal, as they embody the natural rates of growth observed in real-world systems where initial conditions have significant impacts, tapering off as additional changes accumulate.
**Significance of Natural Logarithms in Modeling**- In our exercise, the natural logarithm \(\ln p\) is used, helping to model the average walking speed's natural growth relative to urban population size.- The model’s form \(v(p) = 0.37 \ln p + 0.05\) uses \(\ln p\) to express how initially, for smaller populations, changes in population size have a more substantial effect on walking speed. As a result, it captures the decreasing return on the walking speed increase as the population grows.
Utilizing natural logarithms provides a more realistic portrayal, as they embody the natural rates of growth observed in real-world systems where initial conditions have significant impacts, tapering off as additional changes accumulate.
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