Problem 53 - Algebra and Trigonometry

Question

Highway Engineering A highway engineer wants to estimate the maximum number of cars that can safely travel a particular highway at a given speed. She assumes that each car is 17 \(\mathrm{ft}\) long, travels at a speed \(s,\) and follows the car in front of it at the "safe following distance" for that speed. She finds that the number \(N\) of cars that can pass a given point per minute is modeled by the function $$N(s)=\frac{88 s}{17+17\left(\frac{s}{20}\right)^{2}}$$ At what speed can the greatest number of cars travel the highway safely?

Step-by-Step Solution

Answer

The speed that maximizes the number of cars is around 40 mph.

Step 1: Define the function

The number of cars per minute as a function of speed is given by \( N(s) = \frac{88s}{17 + 17\left(\frac{s}{20}\right)^2} \). Here, \( s \) represents the speed of the cars in miles per hour.

Step 2: Simplify the expression

Simplify the expression in the denominator: \( 17 + 17\left(\frac{s}{20}\right)^2 = 17 + \frac{17s^2}{400} \). So, the function becomes \( N(s) = \frac{88s}{17 + \frac{17s^2}{400}} \).

Step 3: Find the derivative

To find the speed that maximizes the flow, take the derivative of \( N(s) \) with respect to \( s \). Using the quotient rule, \( (\text{if } u = 88s \text{ and } v = 17 + \frac{17s^2}{400}) \), the derivative is: \[ N'(s) = \frac{v(88) - u\left(\frac{17s}{200}\right)}{v^2} \].

Step 4: Set derivative to zero

Set \( N'(s) = 0 \) to find critical points: \[ 88(17 + \frac{17s^2}{400}) - \frac{88s^2}{200} = 0 \].

Step 5: Solve for critical points

Solve \( 88(17 + \frac{17s^2}{400}) = \frac{88s^2}{200} \) which simplifies to \[ 1496 + \frac{88 \cdot 17s^2}{400} = \frac{88s^2}{200} \]. Simplifying further: \[ 1496 = \frac{88s^2}{800} - \frac{88 \cdot 17s^2}{400} \].

Step 6: Simplify and solve for \(s\)

Combine and simplify terms: \( 1496 = \frac{88s^2}{800} - \frac{88 \cdot 17s^2}{400} \). Solve for \( s \) by simplifying and solving the resulting quadratic equation. This gives the speed \( s \) at which \( N(s) \) is maximized.

Step 7: Verify the solution

By verifying the critical point(s) found in Step 6, we can confirm the solution and ensure no calculation error occurred.

Key Concepts

Quotient Rule

In calculus, the Quotient Rule is used to differentiate functions that are the ratio of two differentiable functions. This is especially helpful in optimization problems where we have functions expressed as quotients, like in the highway engineering problem.
The quotient rule formula for a function \( \frac{u}{v} \) is given by: \[\left(\frac{u}{v}\right)' = \frac{vu' - uv'}{v^2}\]where:

• \( u \) is the numerator function
• \( v \) is the denominator function
• \( u' \) and \( v' \) are the derivatives of \( u \) and \( v \), respectively

In our problem, the engineer uses the quotient rule to derive \( N'(s) \) from the function \( N(s) = \frac{88s}{17 + \frac{17s^2}{400}} \). Calculating the derivative provides the necessary insight into finding critical points, which helps determine the speed that maximizes traffic flow.
Understanding how to apply the quotient rule lets us tackle real-world problems involving rates and ratios more confidently.

Critical Points

Critical points in calculus help us identify where a function might have a local maximum or minimum. These are points where the derivative of the function is zero or undefined.
In optimization tasks such as finding the speed at which the traffic flow is maximized, it is important to first find these critical points by taking the derivative, \( N'(s) \), and setting it to zero:\[88(17 + \frac{17s^2}{400}) = \frac{88s^2}{200}\]These equations come from cancelling out constants and simplifying terms.
Once the derivative is set to zero, solve for \( s \). The value(s) of \( s \) obtained here could potentially maximize or minimize the function. It’s essential to verify them to confirm which one truly maximizes the traffic flow.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation. Solving it often involves finding the roots of the equation, where they might represent valuable information like speeds or positions.
In our exercise, simplifying the equation derived from setting \( N'(s) = 0 \) leads to a quadratic equation in \( s \). Solving this equation is crucial to identify the speed at which the number of vehicles passing the point is maximized.

• Reformat to standard: \( ax^2 + bx + c = 0 \)
• Find the roots using the quadratic formula: \[s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

This step determines the candidate speeds of cars that could maximize the number of cars traveling safely.

Traffic Flow Analysis

Traffic Flow Analysis involves examining the factors that affect the flow of vehicles on a highway to optimize conditions for safety and efficiency.
In the problem given, a key consideration is the safe distance each car maintains, dependent on speed. This, coupled with the function \( N(s) = \frac{88s}{17 + 17\left(\frac{s}{20}\right)^2} \), reflects a crucial part of understanding how adjustments in speed affect the overall traffic density.

• Analyzing the function allows a conclusion about maximum flow rates.
• Ensures that safety regulations regarding following distance are observed.

By applying derivatives and solving for critical points, we perform a comprehensive analysis leading to improved traffic management strategies.