Problem 70 - Algebra and Trigonometry with Analytic Geometry

Question

Cars are crossing a bridge that is 1 mile long. Each car is 12 feet long and is required to stay a distance of at least \(d\) feet from the car in front of it (see figure). (a) Show that the largest number of cars that can be on the bridge at one time is \(\llbracket 5280 /(12+d) \rrbracket\), where \(\llbracket \rrbracket\) denotes the greatest integer function. (b) If the velocity of each car is \(v \mathrm{mi} / \mathrm{hr}\), show that the maximum traffic flow rate \(F\) (in cars/hr) is given by \(F=\llbracket 5280 v /(12+d) \rrbracket\).

Step-by-Step Solution

Answer

The largest number of cars is \( \llbracket 5280/(12+d) \rrbracket \); the maximum traffic flow rate is \( \llbracket 5280v/(12+d) \rrbracket \).

Step 1: Calculate Total Car Space on Bridge

The total length of the bridge is 5280 feet (1 mile). Each car takes up 12 feet, and between each car, there must be at least a distance of \(d\) feet. Therefore, the space for each car and the gap following it is \(12 + d\) feet.

Step 2: Express Number of Cars as an Equation

To find the maximum number of cars on the bridge, divide the total bridge length by the space occupied by each car and the gap: \( \text{Number of Cars} = \frac{5280}{12 + d} \).

Step 3: Apply the Greatest Integer Function

Since we can only accommodate a whole number of cars, apply the greatest integer function: \( \llbracket \frac{5280}{12 + d} \rrbracket \). This gives the largest integer less than or equal to the calculated number of cars.

Step 4: Calculate Maximum Traffic Flow Rate

For part (b), consider the velocity \( v \) in miles per hour. The rate of crossing cars is \( v \) miles per hour, so in terms of feet per hour, it's \( 5280v \) feet per hour. The number of cars that can cross per hour is \( \llbracket \frac{5280v}{12 + d} \rrbracket \).

Key Concepts

Greatest Integer Function

The greatest integer function is a mathematical tool used to round a given number down to the nearest whole number. It is represented by the double bracket symbol: \( \llbracket x \rrbracket \). This function is particularly useful when dealing with situations where only whole units are meaningful. In our exercise, the number of cars on a bridge must be a whole number, because you can't have a fraction of a car.

• If you calculate that 5.8 cars could fit on the bridge, the greatest integer function would round this down to 5 cars.
• This ensures that we only count complete cars fully fitting within a given distance or space.

When dividing space on a bridge by the combined length of a car and the mandatory gap behind it, the result isn't always a whole number. The greatest integer function comes in handy here, ensuring we use the largest possible whole number without exceeding the available space.

Velocity

Velocity refers to the speed of something in a given direction. In this context, it relates to the speed at which vehicles travel across the bridge, measured in miles per hour \((mi/hr)\). Understanding velocity is crucial because it affects the flow rate of cars crossing the bridge.

• Velocity, or speed, determines how fast cars can complete crossing the bridge.
• It directly impacts the maximum number of cars that can traverse the span within a specific time frame, influencing overall traffic flow rate.

For instance, if a car travels at a velocity of \(60 \ mi/hr\), this translates into \(5280 \times 60 \) feet per hour. If more cars can cross the bridge quicker thanks to higher speeds, the flow rate, which considers the number of cars passing a point per hour, is higher. It’s important to keep the speed within safe limits to prevent accidents and ensure smooth traffic passage.

Distance Between Cars

The distance between cars, denoted as \( d \) in the problem, represents a safety buffer that needs to be maintained on the bridge. Ensuring a minimum distance is crucial to avoid collisions, especially at the speeds described by their velocity.

• The total space occupied by each car includes the car's length and this safety gap.
• This influences how many cars can fit on the bridge at once and, indirectly, the maximum traffic flow rate.

When calculating the number of cars on the bridge, this distance manipulates the total effective length covered by each car. This means each car, in reality, needs the space of \(12 + d\) feet. Hence, ensuring a safe distance helps manage traffic better and keeps vehicles from being too close to one another, reducing the risk of a potential chain reaction in case of sudden stops.