Problem 60 - Single Variable Calculus

Question

Opened in the late \(1950 \mathrm{~s}\), the Central Artery in downtown Boston was designed to move 75,000 vehicles per day. Suppose that the average speed of traffic flow \(S\) in miles per hour is related to the number of vehicles \(x\) (in thousands) moved per day by the equation $$S=-0.00075 x^{2}+67.5 \quad 50

Step-by-Step Solution

Answer

The rate of change of the average traffic flow speed when there are 100,000 vehicles moved per day is \(-0.15\) miles per hour per thousand vehicles, and when there are 200,000 vehicles moved per day, it is \(-0.3\) miles per hour per thousand vehicles.

Step 1: Write the given function

We have been given the function relating the average traffic flow speed, S, and the number of vehicles, x, as: \[S = -0.00075x^2 + 67.5 \quad {where} \quad 50

Step 2: Differentiate the function with respect to x

To find the rate of change of the average traffic flow speed with respect to the number of vehicles, we need to differentiate the function with respect to x. So, we have: \[\frac{dS}{dx} = \frac{d(-0.00075x^2 + 67.5)}{dx}\] Using the power rule, we find the derivative to be: \[\frac{dS}{dx} = -0.0015x\]

Step 3: Find the rate of change when x is 100,000 vehicles

We are asked to find the rate of change of the average traffic flow speed when 100,000 vehicles are moved per day, which means x = 100. Using the derivative we found in the previous step, we have: \[\frac{dS}{dx} = -0.0015(100)\] Calculating this expression, we get: \[\frac{dS}{dx} = -0.15\] So, the rate of change of the average traffic flow speed when there are 100,000 vehicles moved per day is -0.15 miles per hour per thousand vehicles.

Step 4: Find the rate of change when x is 200,000 vehicles

We are also asked to find the rate of change of the average traffic flow speed when 200,000 vehicles are moved per day, which means x = 200. Using the derivative we found earlier, we have: \[\frac{dS}{dx} = -0.0015(200)\] Calculating this expression, we get: \[\frac{dS}{dx} = -0.3\] So, the rate of change of the average traffic flow speed when there are 200,000 vehicles moved per day is -0.3 miles per hour per thousand vehicles.

Key Concepts

Differentiation in Calculus

Differentiation is a fundamental concept in calculus that deals with the rate at which things change. The process allows us to calculate the instantaneous rate of change of a function with respect to one of its variables. This is incredibly useful in various fields as it helps understand how a small change in one quantity results in a change in another. In the traffic flow example, differentiation helps us determine how the average speed of traffic, denoted as 'S', varies with the number of vehicles 'x' on the road. By finding the derivative of the speed with respect to the number of vehicles, we can predict changes in traffic condition based on the vehicle count.

Power Rule of Derivatives

The power rule is one of the most basic and powerful techniques in differentiation. It states that if you have a function of the form \(x^n\), where \(n\) is any real number, the derivative of that function is \(nx^{n-1}\). Applying this rule makes calculating derivatives straightforward. For our traffic example, the function for speed in terms of vehicles is a quadratic equation, and the power rule simplifies the process of finding its derivative. This rule is essential and especially helpful when dealing with polynomials, as seen with the traffic speed equation \(S = -0.00075x^2 + 67.5\).

Traffic Flow and Calculus

Calculus offers a way to model and analyze the patterns of traffic flow. Using calculus, specifically differentiation, we can model traffic behavior through mathematical functions. The function \(S = -0.00075x^2 + 67.5\) is an example of this, relating the number of vehicles \(x\) to the average speed \(S\). From there, we can derive the rate of change of speed with respect to the number of vehicles, which is vital for traffic engineers as it informs how traffic speed decreases as volume increases. Insights gained from calculus can be used to design better roads, manage traffic flow, and improve overall transportation systems.

Applying Calculus to Real-World Problems

The use of calculus extends beyond academic exercises and is crucial for solving real-world problems. By applying differentiation, we can predict and control various situations. In the case of traffic flow, knowing the rate of change assists in urban planning and infrastructure development. It guides decisions on speed limits, road capacities, and can even influence the timing of traffic lights. Calculus enables us to take a proactive approach to common issues like traffic congestion by providing quantitative data that can be used to implement practical solutions. Understanding these mathematical principles is key for professionals in fields ranging from engineering to economics.

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Key Concepts

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Differentiation is a fundamental concept in calculus that deals with the rate at which things change. The process allows us to calculate the instantaneous rate of change of a function with respect to one of its variables. This is incredibly useful in various fields as it helps understand how a small change in one quantity results in a change in another. In the traffic flow example, differentiation helps us determine how the average speed of traffic, denoted as 'S', varies with the number of vehicles 'x' on the road. By finding the derivative of the speed with respect to the number of vehicles, we can predict changes in traffic condition based on the vehicle count.

Power Rule of Derivatives

The power rule is one of the most basic and powerful techniques in differentiation. It states that if you have a function of the form \(x^n\), where \(n\) is any real number, the derivative of that function is \(nx^{n-1}\). Applying this rule makes calculating derivatives straightforward. For our traffic example, the function for speed in terms of vehicles is a quadratic equation, and the power rule simplifies the process of finding its derivative. This rule is essential and especially helpful when dealing with polynomials, as seen with the traffic speed equation \(S = -0.00075x^2 + 67.5\).

Traffic Flow and Calculus

Calculus offers a way to model and analyze the patterns of traffic flow. Using calculus, specifically differentiation, we can model traffic behavior through mathematical functions. The function \(S = -0.00075x^2 + 67.5\) is an example of this, relating the number of vehicles \(x\) to the average speed \(S\). From there, we can derive the rate of change of speed with respect to the number of vehicles, which is vital for traffic engineers as it informs how traffic speed decreases as volume increases. Insights gained from calculus can be used to design better roads, manage traffic flow, and improve overall transportation systems.

Applying Calculus to Real-World Problems

The use of calculus extends beyond academic exercises and is crucial for solving real-world problems. By applying differentiation, we can predict and control various situations. In the case of traffic flow, knowing the rate of change assists in urban planning and infrastructure development. It guides decisions on speed limits, road capacities, and can even influence the timing of traffic lights. Calculus enables us to take a proactive approach to common issues like traffic congestion by providing quantitative data that can be used to implement practical solutions. Understanding these mathematical principles is key for professionals in fields ranging from engineering to economics.