Problem 67 - 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 is approximately 40 mph.

Step 1: Understand the Problem

We need to find the speed \( s \) that maximizes the number of cars \( N(s) \) safely passing a point on the highway. We are given the function \( N(s) = \frac{88s}{17 + 17(\frac{s}{20})^2} \).

Step 2: Differentiate the Function

To find the maximum, we take the derivative of \( N(s) \) with respect to \( s \). Let's find \( N'(s) \) using the quotient rule.

Step 3: Apply the Quotient Rule

The quotient rule states that if \( f(x) = \frac{g(x)}{h(x)} \), then \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \). Apply this to \( N(s) = \frac{88s}{17 + 17(\frac{s}{20})^2} \) to find \( N'(s) \).

Step 4: Implement Derivative Calculation

Calculate the derivative of the numerator, \( g(s) = 88s \), which is \( g'(s) = 88 \). The denominator, \( h(s) = 17 + \frac{17s^2}{400} \), has a derivative of \( h'(s) = \frac{17s}{200} \). Substitute these into the quotient rule expression.

Step 5: Solve for Critical Points

Set \( N'(s) = 0 \) to find critical points. This involves solving \( 88(17 + 17(\frac{s}{20})^2) - 88s \frac{17s}{200} = 0 \).

Step 6: Simplify and Solve Quadratic Equation

Simplify the equation \( 88 \times 17 + 74.2s^2 - \frac{1496s^2}{200} = 0 \) and solve for \( s \). This reduces to \( 88 \times 17 = \frac{1496s^2}{200} \). Solve for \( s \).

Step 7: Verify Maximum

Use the second derivative test or analyze \( N'(s) \) sign changes around the critical points to ensure that the solution is indeed a maximum.

Step 8: Calculate Numerical Value

Once the critical point is confirmed, calculate the exact value of \( s \). After solving the resulting equation, the speed \( s \) is around \(40 \text{ mph} \).

Key Concepts

Quotient Rule

The quotient rule is a key tool in calculus used to find the derivative of a function that is the division of two differentiable functions. If you have a function of the form \( f(x) = \frac{g(x)}{h(x)} \), where both \( g(x) \) and \( h(x) \) are differentiable, then the quotient rule provides a way to express the derivative \( f'(x) \). The rule states:

• \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \)

This formula calculates the rate of change of the function by considering both the rates of change of the numerator \( g(x) \) and the denominator \( h(x) \), as well as their values themselves. It is critical when dealing with ratios of polynomials or rational functions. Applying the quotient rule carefully helps in determining where a function's value is increasing or decreasing.

Derivative

Derivatives are a fundamental concept in calculus that measure how a function changes as its input changes. Think of a derivative as the slope of a function at any given point, providing the rate at which the function's output is changing.

• For a function \( f(x) \), the derivative \( f'(x) \) describes its rate of change.
• A derivative can be thought of as an "instantaneous rate of change," similar to how speed is the instantaneous rate of change of position.

In the context of the given exercise, finding the derivative \( N'(s) \) of the function \( N(s) \) with respect to speed \( s \) helps identify the speed at which cars can safely pass with optimal efficiency. Calculating derivatives requires careful application of rules like the power rule, product rule, and particularly in this case, the quotient rule.

Critical Points

Critical points occur where the derivative of a function equals zero or does not exist. These points are crucial because they can signal where a function's output may have a maximum or minimum value.

• To find critical points, set the derivative equal to zero and solve for the input variable.
• Analyze these points further to determine if they indicate local or global maxima or minima.

In the exercise, solving for the critical points involves setting \( N'(s) = 0 \) and solving for the speed \( s \) values that meet this condition. Checking these critical points verifies where the most cars can safely pass at a specific speed, maximizing traffic flow efficiency.

Safe Following Distance

Safe following distance is a practical concept in road safety that maintains enough space between vehicles to allow for sudden stops or emergencies. It ensures that each vehicle has adequate time to react and stop safely.

• This distance increases with speed since higher speeds require more time to slow down safely.
• Traffic engineers use mathematical models to balance safety with road efficiency.

In the context of the exercise, the safe following distance is factored into the function \( N(s) \) that estimates the number of cars passing a point at a particular speed. It reflects the space each car needs to safely follow another, which impacts the overall capacity of the highway lanes. Understanding this dynamics helps engineers optimize speed limits and traffic flow.