Problem 88 - Intermediate Algebra

Question

Solve each problem by writing a variation model. Braking. 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 for a vehicle traveling at 50 mph.

Step 1: Understanding the Problem

The problem states that stopping distance varies directly with the square of speed. This implies a direct variation model: if speed increases, the stopping distance increases with the square of the speed. We'll use the relation between stopping distance and speed to form an equation.

Step 2: Setting Up the Direct Variation Equation

The relationship can be expressed as a variation model: \( d = k v^2 \), where \( d \) is the stopping distance, \( v \) is the speed, and \( k \) is the variation constant. We need to determine \( k \) using the information that a speed of 20 mph results in a stopping distance of 24 feet.

Step 3: Calculating the Variation Constant

Substitute the known values into the equation \( d = k v^2 \): \( 24 = k imes 20^2 \). Simplifying this gives \( 24 = 400k \). Solving for \( k \), we find \( k = \frac{24}{400} = 0.06 \).

Step 4: Finding the Stopping Distance for 50 mph

Now that we know \( k = 0.06 \), we can find the stopping distance for a speed of 50 mph. Substitute 50 for \( v \) in the model: \( d = 0.06 imes 50^2 \).

Step 5: Simplifying the Equation

Calculate \( 50^2 = 2500 \). Substitute back into the equation to find \( d = 0.06 imes 2500 \).

Step 6: Final Calculation and Result

Calculate \( d = 0.06 imes 2500 = 150 \). Thus, the stopping distance for a vehicle traveling at 50 mph is 150 feet.

Key Concepts

Direct Variation Explained

Direct variation is a mathematical concept where one quantity increases or decreases proportionally with another. This relationship can be expressed with the formula \( y = kx \), where \( y \) and \( x \) are two quantities and \( k \) is the constant of variation. In real life, if something varies directly, as one factor goes up, the other goes up, too. For example, the stopping distance of a car varies directly with the square of its speed. This means if you double the speed, the stopping distance goes up by four times because you're squaring the speed.Understanding direct variation helps in predicting outcomes based on changes in variables that are directly connected. It is important to identify the constant \( k \), which shows how strong or weak the relationship is between the variables.

The Science of Stopping Distance

Stopping distance is the total distance a vehicle travels before it completely stops after the brakes are applied. In our problem, it depends on the speed of the vehicle squared. Here's why: - **Speed:** The initial speed of the vehicle plays a huge role, as faster vehicles require more distance to stop due to greater momentum. - **Square of the Speed:** This means small increases in speed significantly increase the stopping distance, which is crucial for safety considerations. Calculating stopping distances can help drivers understand how much space they need on the road to stop safely, especially at higher speeds. This is why speed limits are crucial for safe driving.

Speed and Distance Relationship

The relationship between speed and stopping distance is crucial in determining how velocity affects the space needed to safely stop a vehicle. - **Direct Relation:** As speed increases, stopping distance increases dramatically. A small increase in speed can lead to a significantly longer stopping distance. - **Practical Implications:** Understanding this relationship helps in setting safe speed limits and in designing safer vehicles. It's also important for drivers as it influences how they drive under various conditions. For example, if a car's speed increases from 20 mph to 50 mph, the stopping distance doesn’t just double — due to the squared factor in the equation, it becomes much longer. This illustrates the non-linear relationship between speed and stopping distance, emphasizing caution at higher speeds.