Problem 38 - Elementary and Intermediate Algebra

Question

Solve each problem by writing a variation model. Suppose the distance that a vehicle travels after its brakes have been applied varies directly as the square of the speed at which it was traveling. If the stopping distance for such a vehicle going 20 mph is 24 feet, what is the stopping distance for the vehicle traveling at 50 mph?

Step-by-Step Solution

Answer

The stopping distance is 150 feet when traveling at 50 mph.

Step 1: Understand Direct Variation Model

In this problem, the stopping distance (\( d \)) varies directly as the square of the speed (\( s \)). This can be expressed by the equation:\[d = k \cdot s^2\]where \( k \) is the constant of variation.

Step 2: Find the Constant of Variation

We are given that at a speed of 20 mph, the stopping distance is 24 feet. Substitute these values into the equation \( d = k \cdot s^2 \) to solve for \( k \).\[24 = k \cdot (20)^2\]Simplify and solve for \( k \):\[24 = k \cdot 400\]So,\[k = \frac{24}{400} = \frac{3}{50}\]

Step 3: Apply the Constant to Find New Stopping Distance

Now, use \( k = \frac{3}{50} \) to find the stopping distance when the speed is 50 mph. Substitute \( s = 50 \) into\[d = k \cdot s^2\]\[d = \frac{3}{50} \cdot (50)^2\]\[d = \frac{3}{50} \cdot 2500\]Simplify:\[d = \frac{7500}{50} = 150\]

Step 4: Conclusion

The stopping distance for the vehicle traveling at 50 mph is 150 feet.

Key Concepts

Direct Variation

In many real-world scenarios, variables can be directly related to each other by a constant proportion. This type of relationship is known as a "direct variation." For example, if one variable increases, the other variable increases at a constant rate as well.

The equation that represents direct variation typically looks like this:

• \( y = k imes x \)
• where \( y \) and \( x \) are the two variables, and \( k \) is the constant of variation.

Understanding direct variation makes it easier to predict how changes in one element will affect another. In the context of our exercise, the stopping distance (**d**) of a vehicle "varies directly as" the square of the speed (**s**), meaning that as the speed changes, the stopping distance changes proportionally based on this square relationship.

This direct variation equation can be expressed as:

• \( d = k imes s^2 \) where \( k \) is the constant of variation.

Stopping Distance

"Stopping distance" is the distance a vehicle travels from the point where its brakes are fully applied to when it comes to a complete stop. It's crucial to understand that stopping distance can be affected by many factors, such as road conditions, braking systems, and importantly, speed.

In the given problem, the stopping distance increases with the square of the speed. For example, if a vehicle's speed doubles, its stopping distance actually quadruples due to the square relation in the formula \( d = k imes s^2 \).

Why is this important? Knowing how stopping distance is computed makes it easier for drivers to understand the risks involved at higher speeds. This knowledge can be especially critical for making safe driving decisions and for engineers in designing better braking systems.

Speed and Distance Relationship

The relationship between speed and the distance a vehicle travels while stopping is quite intuitive once you break it down. At its core, the speed at which you're traveling directly influences how far it will take to come to a stop. However, this relationship is not linear; it's quadratic due to the nature of the equation \( d = k \times s^2 \).

For example:

• At 20 mph, if the stopping distance is 24 feet, doubling the speed to 40 mph doesn't just double the stopping distance. It actually increases it to four times the original distance. This is because the speed squared (40\(^2\) is four times 20\(^2\)) results in a larger stopping distance.

Understanding this quadratic relationship is vital. It underscores the necessity of speed regulations and the prominent role speed plays in traffic safety. Drivers should always be aware that even a modest increase in speed can significantly increase the distance required to stop safely. This knowledge is not just theoretical— it's practical for everyday driving.