Problem 36 - Algebra and Trigonometry

Question

Stopping Distance The stopping distance \(D\) of a car after the brakes have been applied varies directly as the square of the speed \(s\) . A certain car traveling at 50 \(\mathrm{mi} / \mathrm{h}\) can stop in 240 \(\mathrm{ft}\) . What is the maximum speed it can be traveling if it needs to stop in 160 \(\mathrm{ft}\) ?

Step-by-Step Solution

Answer

The maximum speed is approximately 25.82 mph.

Step 1: Understand the Relationship

The stopping distance \(D\) varies directly as the square of the speed \(s\), which means we can express this relationship as \(D = k \cdot s^2\), where \(k\) is a constant of proportionality.

Step 2: Find the Constant of Proportionality

We know that when \(s = 50\ \text{mi/h}\), the stopping distance \(D = 240\ \text{ft}\). Plugging these values into the equation \(240 = k \cdot (50)^2\), we solve for \(k\). Thus, \(k = \frac{240}{2500} = \frac{12}{125}\).

Step 3: Use the Constant to Find Maximum Speed

We need to find the speed at which the stopping distance is 160 ft. Using \(D = k \cdot s^2\) with \(D = 160\), we have \(160 = \frac{12}{125} \cdot s^2\). Solving for \(s^2\), we get \(s^2 = \frac{160 \times 125}{12} = \frac{2000}{3}\).

Step 4: Calculate the Speed

To find \(s\), take the square root of \(\frac{2000}{3}\). Thus, \(s = \sqrt{\frac{2000}{3}} \approx 25.82 \ \text{mi/h}\).

Key Concepts

Stopping Distance

When driving, the stopping distance of a vehicle plays a crucial role in road safety. Stopping distance is the distance a car travels after the brakes are applied and comes to a complete stop. This concept is vital because it determines the time a driver has to avoid potential hazards.

The factor influencing stopping distance the most is speed. Specifically, the stopping distance varies directly as the square of the car's speed. This means as a vehicle's speed increases, the stopping distance does not just increase proportionately; it grows at an accelerated rate.

At higher speeds:

• Stopping distances are longer, which means more space is needed to stop the car safely.
• This relationship emphasizes the importance of adjusting speed according to driving conditions to maintain safety.

Proportional Relationships

Proportional relationships describe situations where two quantities increase or decrease at the same rate. In the case of stopping distance and speed, we use a direct variation model. This is represented mathematically by the formula:

\[ D = k \cdot s^2 \]

Here:

• \(D\) is the stopping distance,
• \(s\) is the speed,
• \(k\) is the constant of proportionality, defining how the two quantities vary together.

A direct variation means that if you know the constant \(k\), you can determine one variable if the other is known. This is practical for drivers to estimate their stopping distances based on their speed, given a specific constant for their vehicle and road conditions.

In our problem, knowing the relationship helps calculate how fast a car can safely travel and still stop within a certain distance. Adjusting speed and calculating stopping distances using this proportional relationship can enhance understanding and application of safe driving practices.

Quadratic Functions

The formula for stopping distance reveals quadratic nature because it involves the square of speed, \(s^2\). Quadratic functions are essential in mathematics since they model a variety of real-world situations where a relationship forms a parabola when graphed. They help understand complex behaviors where one quantity depends on the square of another.

This quadratic relationship in stopping distance indicates:

• The increase in stopping distance's concave upward curve is typical of quadratic growth.
• Changes in speed have a more pronounced effect due to it being squared—a 10% increase in speed results in over a 20% increase in stopping distance.

Understanding quadratic functions allows us to solve for unknown variables effectively. In our example problem, it clarifies how to determine the highest speed possible that still allows for a stipulated stopping distance, using the properties of quadratic equations. The quadratic equation helps us find viable solutions for speed that adhere to safety requirements, highlighting the practical application of quadratic functions.