Problem 80 - Elementary Algebra

Question

Assuming dry road conditions and average reaction times, the safe stopping distance, \(d\) in feet of an average car is given using the formula \(d=12002+v\), where v represents the speed of the car in miles per hour. For each problem below, given the stopping distance, determine the safe speed. $$ 120 \text { feet } $$

Step-by-Step Solution

Answer

There's an error in the setup as the formula used doesn't seem correct.

Step 1: Understand the Problem

We are given the formula for the safe stopping distance of a car: \( d = 12002 + v \), where \( d \) is the stopping distance in feet and \( v \) is the speed in miles per hour. We need to find the speed \( v \) when the stopping distance \( d \) is 120 feet.

Step 2: Set Up the Equation

To find the speed \( v \), we substitute \( d = 120 \) feet into the formula: \( 120 = 12002 + v \).

Step 3: Solve for Speed \( v \)

Now, solve the equation for \( v \). Subtract \( 12002 \) from both sides to isolate \( v \):\[ 120 - 12002 = v \]Calculate the left side to find \( v \).

Step 4: Calculate the Value

Calculate \( 120 - 12002 \):\[ v = -11882 \]

Step 5: Interpret the Result

The negative value indicates an error in the initial setup as speed cannot be negative. Check the problem formula, as it seems there would be a mistake because this formula results in unrealistic speeds.

Key Concepts

Stopping Distance

In driving, stopping distance refers to the total distance a vehicle travels before it comes to a complete stop after the driver perceives a need to brake. This distance is crucial for understanding safe driving practices. It's typically composed of two parts:

• **Reaction Distance:** The distance covered during the driver's reaction time.
• **Braking Distance:** The distance from when the brakes are applied until the vehicle stops.

In the given exercise, the formula initially intended to represent this concept was incorrectly stated. Generally, the accurate calculation involves both components, tying into factors such as road conditions and vehicle speed. Under typical conditions, the formula would look more like:
\[d = ext{(reaction time component)} + ext{(braking distance component)},\]where each component accurately accounts for speed and other variables. However, due to a typo in the original exercise, it presented an added constant instead of a multiplicative factor, leading to the unexpected result.

Reaction Time in Driving

Reaction time in driving is the time lapse between the driver's realization that they need to stop and the moment they press the brake pedal. This is a critical factor in determining stopping distance and is influenced by several factors, including:

• Driver's alertness and fatigue.
• Visibility and road conditions.
• Vehicle performance (e.g., brake system responsiveness).

Reaction time can vary from about 0.75 to 1.5 seconds for most drivers under standard conditions. It's important in auto safety calculations because a delay in reaction increases the distance a vehicle travels before braking, thereby extending the overall stopping distance. Understanding this helps drivers maintain safe distances from other vehicles, adjust speeds, and anticipate stops more effectively.

Negative Results in Equations

Negative results in algebraic equations, particularly when determining values like speed or distance, usually indicate an error in the initial setup or calculation. This occurs because in practical measurements, values such as distance and speed cannot be negative.
In our exercise, the result \[v = -11882\]was due to an erroneous equation. Proper algebraic handling should reflect plausible real-world scenarios. Here's how you can approach such situations:

• Recheck the equation and make sure all terms logically relate to the scenario described.
• Verify constants and variables align with the problem's physical context.
• Consult alternative documentation or resources to ensure formula correctness.

This analysis helps correct any missteps and reinforces the necessity of logical consistency between mathematical results and real-world interpretations. In this instance, recognizing the error prompts a reassessment of the problem's structure, guiding one back to a correct formulation.