Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Speeding Fines Suppose that speeding fines are determined by \(y=10(x-65)+50, x>65,\) where \(y\) is the cost in dollars of the fine if a person is caught driving \(x\) miles per hour. (a) How much is the fine for driving 76 mph? (b) While balancing the checkbook, Johnny found a check that his wife Gwen had written to the Department of Motor Vehicles for a speeding fine. The check was written for \(\$ 100 .\) How fast was Gwen driving? (c) At what whole-number speed are tickets first given? (d) For what speeds is the fine greater than \(\$ 200 ?\)

Step-by-Step Solution

Answer

(a) $160; (b) 70 mph; (c) 66 mph; (d) Over 80 mph.

Step 1: Calculate fine for 76 mph

For part (a), substitute \(x = 76\) into the formula \(y = 10(x-65) + 50 \): \[ y = 10(76-65) + 50 \] \[ y = 10(11) + 50 \] \[ y = 110 + 50 \] \[ y = 160 \] Thus, the fine for driving 76 mph is \( \$160 \).

Step 2: Find speed for $100 fine

For part (b), set \(y = 100\) and solve for \(x\): \[ 100 = 10(x-65) + 50 \] Subtract 50 from both sides:\[ 50 = 10(x-65) \] Divide both sides by 10:\[ 5 = x - 65 \] Add 65 to both sides:\[ x = 70 \] Gwen was driving at 70 mph.

Step 3: Determine starting speed for fines

For part (c), set \(x = 65\) in the condition of the function. Since speeding fines start when \(x > 65\), a ticket is first given at 66 mph.

Step 4: Identify speeds with fines over $200

For part (d), set \(y > 200\) and solve for \(x\): \[ 10(x-65) + 50 > 200 \] Subtract 50 from both sides:\[ 10(x-65) > 150 \] Divide by 10:\[ x-65 > 15 \] Add 65 to both sides:\[ x > 80 \] Fines are greater than \( \$200 \) for speeds over 80 mph.

Key Concepts

Speeding Fines Calculation

Calculating speeding fines can often confuse students due to the combination of a linear formula and specific speed thresholds. In real-life scenarios, as in this problem, the amount you are fined is directly linked to how much you exceed a set speed limit.
The formula given in the exercise - begins with a base fine - adds an additional amount per mile over the limit For an example, if one exceeds the limit by 11 miles, the formula calculates the fine by multiplying this excess by 10, then adding 50. Recognizing this relationship is key to solving various parts of the problem related to speeding fines.

Linear Equations

Linear equations are vital in identifying a direct relationship between two variables, like speed and fines in our example. The basic form of a linear equation is depicted in the formula used for speeding fines: \(y=10(x-65)+50\).
Here’s what each component means: - "x" represents the speed of the car. - "y" corresponds to the fine amount.The equation shows how fines increase with speed. As speed increases by one mile per hour over 65 mph, the fine increases by 10 dollars. Understanding this basic linearity is crucial in the financial aspects of real-world speeding fines.

Inequalities in Mathematics

Inequalities extend the idea of equations by introducing concepts such as greater than or less than. In this exercise, you analyze when a fine might exceed a certain amount, specifically in parts (d), where it asks for fines greater than \(\\(200\).
To solve: - Begin by using the formula \(y > 200\).- Applying algebraic manipulation to isolate x on one side.The result suggests speeds resulting in fines above \(\\)200\), specifically speeds over 80 mph. Mastering inequalities allows you to reason beyond exact equations and examine ranges that quantify real-world conditions.

Problem Solving Steps

Solving word problems in precalculus requires a series of logical steps to connect the question to the solution effectively. Here’s how:- **Identify Keywords:** Recognize terms like "fine," "speed," or "greater than."- **Define Variables:** Determine what variable each keyword represents. This helps you set up your problem correctly.- **Formula Application:** Apply the given formulas and constraints. In our example, substitute given values to see the outcome.- **Check Work:** After finding a solution, ensure it makes sense contextually, like with Gwen’s speed, checking if \(\$100\) aligns with her speed.Using these steps systematically ensures you approach problems methodically. Learning this process can simplify even the most complex precalculus problems.