To solve this problem, we need to evaluate the given function for the maximum traffic flow and its rate of change with respect to the street width when $x = 34$ feet. The function is defined as:

$f(x) = \frac{28,548 \cdot (2.364 - \log x)}{x}$

where $x$ is the street width in feet, $\log x$ is the natural logarithm (base $e$), and $f(x)$ represents the maximum traffic flow in vehicles per hour. Here's how to proceed:

1. Finding the Maximum Traffic Flow:

We substitute $x = 34$ into the function:

$f(34) = \frac{28,548 \cdot (2.364 - \log 34)}{34}$

Let's calculate this value step-by-step.

2. Finding the Rate of Change with Respect to $x$:

The rate of change of $f(x)$ with respect to $x$ is found by taking the derivative of $f(x)$, denoted as $f'(x)$. Using the quotient rule for differentiation, we can compute $f'(x)$:

$f'(x) = \frac{d}{dx} \left( \frac{28,548 \cdot (2.364 - \log x)}{x} \right)$

After finding $f'(x)$, we evaluate it at $x = 34$.

Let's perform these calculations.1. Maximum Traffic Flow: The maximum traffic flow for a street width of 34 feet is approximately -976 vehicles per hour (rounded to the nearest integer).

2. Rate of Change: The rate of change of the maximum traffic flow with respect to street width at 34 feet is approximately 4.01 vehicles per hour per foot.

Let me know if you'd like further details or have any questions.

Related Questions:

1. How would the maximum traffic flow change if the street width was 50 feet?
2. What would be the rate of change for a street width of 40 feet?
3. Can you find the critical points for the function $f(x)$?
4. What does a negative maximum traffic flow signify in the context of this problem?
5. How does the maximum traffic flow behave as $x$ becomes very large?

Tip:

To analyze how a function behaves as its input increases or decreases significantly, consider its limit properties to get a general understanding of long-term trends.