To solve the problem, let’s break it down into parts:

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(a) Exponential Model

We are tasked to find an approximate exponential model for the relationship between density $D$ and speed $s$. The general form of an exponential function is:

$D(s) = A \cdot e^{Bs}$

Here $A$ and $B$ are parameters to be determined.

Using the data: [ s = [32, 25, 20, 17, 13] \quad \text{and} \quad D = [34, 53, 74, 88, 102] ]

Take the natural logarithm of both sides to linearize the relationship: $\ln(D) = \ln(A) + Bs$

This converts the exponential relationship into a linear one, which can be fit using least squares regression. I will calculate this for you. numpy as np import pandas as pd from scipy.stats import linregress

Given data

speed = np.array([32, 25, 20, 17, 13]) # s density = np.array([34, 53, 74, 88, 102]) # D

Linearizing the data by taking the natural log of density

ln_density = np.log(density)

Perform linear regression on (speed, ln_density)

slope, intercept, r_value, p_value, std_err = linregress(speed, ln_density)

Extract parameters A and B for the exponential model

A = np.exp(intercept) # A = e^(intercept) B = slope

Round parameters to three decimal places

A_rounded = round(A, 3) B_rounded = round(B, 3)

A_rounded, B_roundedThe approximate exponential model is:

$D(s) = 233.206 \cdot e^{-0.059s}$

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(b) Density at Average Speed of 26 mph

Using the model $D(s) = 233.206 \cdot e^{-0.059s}$, calculate the density when $s = 26$:

$D(26) = 233.206 \cdot e^{-0.059 \cdot 26}$

Let’s compute this. Calculate density for s = 26 speed_26 = 26 density_26 = A * np.exp(B * speed_26)

Round to nearest whole number

density_26_rounded = round(density_26) density_26_roundedThe density of traffic flow when the average speed is 26 miles per hour is:

$D(26) = 50 \, \text{cars/mi}$

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(c) Effect of Increasing Speed by 1 mph

To determine the percentage decrease in density when the speed increases by 1 mph, calculate the relative change using the exponential model. The percentage decrease can be expressed as:

$\text{Percent decrease} = \left( 1 - \frac{D(s+1)}{D(s)} \right) \cdot 100$

Let’s compute the percentage decrease for a 1 mph increase. Calculate densities for s and s+1 density_s = A * np.exp(B * speed_26) # D(s) where s = 26 density_s_plus_1 = A * np.exp(B * (speed_26 + 1)) # D(s+1) where s = 27

Calculate percentage decrease

percent_decrease = (1 - density_s_plus_1 / density_s) * 100

Round to one decimal place

percent_decrease_rounded = round(percent_decrease, 1) percent_decrease_roundedThe density decreases by approximately:

$5.8\% \, \text{for every 1 mph increase in average speed.}$

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Summary:

1. Exponential Model: $D(s) = 233.206 \cdot e^{-0.059s}$
2. Density at 26 mph: $D(26) = 50 \, \text{cars/mi}$
3. Effect of 1 mph Speed Increase: The density decreases by $5.8\%$.

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Would you like additional clarification or further calculations? Here are five related questions to explore:

1. How does the model perform for other speeds (e.g., $s = 30$)?
2. What is the predicted density when $s = 10$ mph?
3. How does the rate of decrease in density change with higher speeds?
4. Could a different functional form (e.g., polynomial) better fit the data?
5. How does the goodness-of-fit (e.g., $R^2$) support the exponential model's accuracy?

Tip: Always validate models with more data points to ensure they generalize well.