Problem 79 - Algebra and Trigonometry

Question

Speeding Tickets In a certain state the maximum speed permitted on freeways is 65 mi/h, and the minimum is 40. The fine F for violating these limits is $15 for every mile above the maximum or below the minimum. (a) Complete the expressions in the following piecewise defined function, where x is the speed at which you are driving. (b) Find F(30), F(50), and F(75). (c) What do your answers in part (b) represent?

Step-by-Step Solution

Answer

F(30) = $150, F(50) = $0, F(75) = $150; these represent fines for speed violations.

Step 1: Define the Problem

We need to find the expression for the fine \( F \) based on the speed \( x \). This requires creating a piecewise function based on the speed limits (maximum of 65 mi/h and minimum of 40 mi/h).

Step 2: Create the Piecewise Function

The piecewise function for the fine \( F(x) \) is defined as follows: \[F(x) = \begin{cases} 15(x - 65), & \text{if } x > 65 \0, & \text{if } 40 \leq x \leq 65 \15(40 - x), & \text{if } x < 40 \end{cases}\]This covers speeding and driving under the minimum speed limits.

Step 3: Calculate F(30)

Using our piecewise function, plug in \( x = 30 \) which is below the minimum speed. Hence, use the formula \( F(x) = 15(40 - x) \):\[ F(30) = 15(40 - 30) = 15 \times 10 = 150 \]The fine for driving at 30 mi/h is $150.

Step 4: Calculate F(50)

For \( x = 50 \) which is within the permitted speed limits, use the formula \( F(x) = 0 \):\[ F(50) = 0 \]There is no fine for driving at 50 mi/h.

Step 5: Calculate F(75)

Using our piecewise function, plug in \( x = 75 \) which is above the maximum speed. Hence, use the formula \( F(x) = 15(x - 65) \):\[ F(75) = 15(75 - 65) = 15 \times 10 = 150 \]The fine for driving at 75 mi/h is $150.

Step 6: Interpret the Results

The results show that when driving under the minimum speed or over the maximum speed, a fine is incurred. Specifically: - At 30 mi/h, a $150 fine applies for being under the minimum speed. - At 50 mi/h, there is no fine since this is within the legal speed limit. - At 75 mi/h, a $150 fine applies for exceeding the maximum speed.

Key Concepts

Speed Limit

The concept of speed limit is fundamental to safe driving on roads and highways. A speed limit defines the maximum and minimum speeds at which vehicles can legally travel. These limits ensure a smooth flow of traffic and enhance safety for drivers, passengers, and pedestrians.
The speed limit is typically indicated by road signs and varies depending on the type of road, location, and current traffic regulations:

• Maximum Speed Limit: The highest speed at which you are legally allowed to drive. For the given exercise, this is 65 mi/h.
• Minimum Speed Limit: The lowest speed you are permitted to maintain on the highway. Here, it is set at 40 mi/h.

Violating these speed limits can lead to traffic penalties such as fines, which are calculated based on the degree of violation.

Fine Calculation

Fines for speed limit violations are not just random numbers; they are calculated according to specific rules, usually determined by local laws and regulations. In the exercise, the fine is determined using a simple rule:
If you go above the maximum speed or below the minimum speed, you incur a fine of $15 for each mile per hour (mi/h) you are over or under the limit. This means:

• Above 65 mi/h: The fine is $15 multiplied by the number of miles over 65.
• Below 40 mi/h: The fine is $15 multiplied by the number of miles under 40.

This structured fine calculation ensures a proportional punishment system, discouraging people from speeding or driving dangerously slow, which can disrupt traffic flow.

Speeding Tickets

Receiving a speeding ticket is a common consequence of violating speed limits. A speeding ticket formally notifies the driver that they have committed a traffic offense. In the context of the exercise:
Speeding tickets serve both as a deterrent and a record of a driver's traffic violations. When you drive faster than the maximum allowable speed, you're more likely to receive a ticket along with a fine:

• In this scenario, driving at 75 mi/h results in a $150 fine.
• Conversely, driving at 30 mi/h also results in a $150 fine due to being under the minimum speed.

This highlights the dual role of tickets in enforcing both maximum and minimum speed regulations.

Mathematical Expression

A mathematical expression is used to calculate the fines associated with speeding and driving below the speed limit. In this case, the expression is part of a piecewise defined function. The established formulas are based on conditional statements that help determine the appropriate fine:

• <(x - 65) * 15> for speeds over 65 mi/h.
• 0 for speeds between 40 and 65 mi/h.
• <(40 - x) * 15> for speeds under 40 mi/h.

By substituting specific speed values into these expressions, you can easily calculate the exact fine amount for any given speed violation. This systematic approach helps in understanding how fine amounts are derived mathematically.

Piecewise Defined Function

A piecewise defined function is a function composed of multiple sub-functions, each of which is applicable to a certain interval of the main function's domain. It is especially useful for defining mathematical rules that have different outcomes based on varying input ranges.
In the speeding tickets exercise, the fine is determined by a piecewise function. This function assigns different sub-functions for different speed intervals:

• For speeds greater than 65 mi/h, the rule is 15(x - 65).
• For speeds between 40 and 65 mi/h, the fine is 0.
• For speeds less than 40 mi/h, the rule is 15(40 - x).

This type of function helps represent real-world situations where conditions change based on certain boundaries, such as speed limits in this example.