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Chapter 3: Problem 11

Fines for a particular speeding ticket are defined by the following piecewise function, where \(s\) is the speed in mph and \(F(s)\) is the fine in dollars. $$ F(s)=\left\\{\begin{array}{ll} 0 & \text { if } \quad s \leq 45 \\ 50+5(s-45) & \text { if } 4565 \end{array}\right. $$ a. What is the implied posted speed limit for this situation? b. Create a table of values for the fine, beginning at \(40 \mathrm{mph}\) and incrementing by 5 -mph steps up to \(80 \mathrm{mph}\), and then \(\operatorname{graph} F(s)\) c. Describe in words how a speeding fine is calculated. d. Explain what \(5,10,\) and 20 in the formulas for the respective sections of the piecewise function represent. e. Find \(F(30), F(57),\) and \(F(67)\). f. Graph \(F(s)\).

Short Answer

Expert verified
1. 45 mph, 2. Table: (see step 2), 3. Fines increase based on the speed range, 4. Rates of fine increase per mph over certain thresholds, 5. F(30)=0, F(57)=120, F(67)=220, 6. Graph: Piecewise linear segments with given slopes

Step by step solution

01

Identify the posted speed limit

The posted speed limit is typically the speed at which there is no fine. Since there is no fine when the speed is less than or equal to 45 mph, the implied posted speed limit is 45 mph.
02

Create a table of values

Begin at 40 mph and create the table with 5-mph increments up to 80 mph. Look at each segment of the piecewise function to determine the fines for each speed:
03

Calculate Fines for Specific Speeds

Use the piecewise function to calculate the fine at each speed value:
04

Evaluate Specific Speeds

Use the piecewise function to find the fines:

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

speeding fines
Speeding fines are often structured to increase with the severity of the offense. In this exercise, a piecewise function defines the fines. The function varies depending on how much the speed exceeds a specific limit. The posted speed limit here is identified as 45 mph. Understanding this is crucial, as any speed up to and including 45 mph incurs no fine.
For speeds between 45 and 55 mph, the fine is computed using the formula: \(F(s) = 50 + 5(s - 45)\). For speeds between 55 and 65 mph, the formula changes to: \(F(s) = 100 + 10(s - 55)\). Fines escalate significantly for speeds above 65 mph, calculated using: \(F(s) = 200 + 20(s - 65)\).
This setup ensures that higher speeds face exponentially higher fines, acting as a deterrent for severe speeding.
function graphing
Graphing the piecewise function provided can aid in visualizing how fines increase with speed. Begin by creating a table of values that include speeds from 40 mph to 80 mph in increments of 5 mph. Calculate the corresponding fines using the given formulas.
For example, at 40 mph, the fine is \(0\). At 50 mph, the fine is determined by substituting into the formula \(50 + 5(5) = 75\). Continue this method until 80 mph.
Once you have the table of values, plot the speeds on the x-axis and the fines on the y-axis. Connect the dots to illustrate the different segments of the piecewise function. These segments will show different slopes, indicating increasing fines with increasing speeds. You will see a step-like function where each segment has a different rate of increase.
algebraic expressions
Algebraic expressions simplify the calculation of speeding fines within different speed ranges. Here, we deal with three unique expressions:
  • eighboring constant rate: \(F(s) = 50 + 5(s - 45)\) for 45 < s 鈮� 55 mph
  • a steeper climb: \(F(s) = 100 + 10(s - 55)\) for 55 < s 鈮� 65 mph
  • an even more aggressive increase: \(F(s) = 200 + 20(s - 65)\) for s > 65 mph
These pieces of the function break down the larger relationship into manageable parts. By substituting specific speeds into these expressions, one can determine the exact fine for any given speed within a segment. These expressions make it easier to calculate values without misunderstanding or error.
step-by-step problem solving
Approaching piecewise functions requires a structured problem-solving strategy. Follow these outlined steps for accuracy:
  1. Identify the speed limit: The speed up to which there is no fine, here it is 45 mph.
  2. Create the table: List speeds starting at 40 mph in 5 mph increments up to 80 mph.
  3. Calculate fines: Use the specific algebraic expression for each speed range. For example, to find the fine at 57 mph, use \(F(s) = 100 + 10(s - 55)\) and solve: \(F(57) = 100 + 10(57 - 55) = 100 + 20 = 120\). Repeat for other speeds.
  4. Graph the function: Plot the calculated fines against the speeds. Join the points for each range segment to visualize the step-like increases.
By methodically following these steps, you'll better understand how the fines are structured and can accurately apply the function to various speeds.

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Most popular questions from this chapter

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