Problem 86 - Algebra A Combined Function

Question

The Penske Racing team has been involved in the Indianapolis 500 for many successful years. Two of its drivers, Helio Castroneves and Ryan Briscoe, placed 4th and 23 rd, respectively, in the 2008 Indianapolis 500 . The track is 2.5 miles long. When traveling at their fastest lap speeds, Briscoe drove 2.492 miles in the same time that Castroneves completed an entire 2.5-mile lap. Castroneves' fastest lap speed was 0.694 mph faster than Briscoe's fastest lap speed. Find each driver's fastest lap speed. Round each speed to the nearest tenth. (Source: Indy Racing League)

Step-by-Step Solution

Answer

Briscoe's fastest lap speed is 216.1 mph, and Castroneves' is 216.8 mph.

Step 1: Define Variables

Let the speed of Briscoe be \( x \) mph. Since Castroneves' speed is 0.694 mph faster, his speed is \( x + 0.694 \) mph.

Step 2: Formula for Time

The time taken to complete a distance is given by the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).

Step 3: Create Equation for Castroneves

Using the time formula, the time for Castroneves to complete a lap is \( \frac{2.5}{x + 0.694} \).

Step 4: Create Equation for Briscoe

The equation for Briscoe completing 2.492 miles is \( \frac{2.492}{x} \).

Step 5: Set Time Equations Equal

Since they take the same time, equate both expressions: \[ \frac{2.5}{x + 0.694} = \frac{2.492}{x} \].

Step 6: Solve Equation for x

Cross-multiply and solve for \( x \):\[ 2.5x = 2.492(x + 0.694) \]. Expand and simplify: \[ 2.5x = 2.492x + 1.729048 \].Subtract \( 2.492x \) from both sides to get: \[ 0.008x = 1.729048 \].Divide both sides by 0.008: \[ x \approx 216.131 \].

Step 7: Find Castroneves' Speed

Use the value of \( x \) to find Castroneves' speed: \( x + 0.694 = 216.131 + 0.694 = 216.825 \).

Step 8: Round to the Nearest Tenth

Round Briscoe's speed to 216.1 mph and Castroneves' speed to 216.8 mph.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that use letters to represent numbers. In this exercise, these equations help us find the fastest lap speeds of two racing drivers. The letters in the equations, referred to as variables, stand for the unknown values we're trying to solve. In our case, we denote Briscoe's speed using the variable \( x \). This allows us to express the relationship between the unknown speeds and the known conditions of the race.
The exercise tells us that Castroneves' speed is 0.694 mph faster than Briscoe's. We incorporate this information into the equation by expressing Castroneves' speed as \( x + 0.694 \). By setting up these equations, we can work through the problem to find the unknown speeds step-by-step.

Problem Solving Steps

Solving a complex problem like this involves several clearly defined steps. Here's a breakdown of how we can approach it:

• Step 1: Define the variables. Identify what each variable represents. In this case, \( x \) represents Ryan Briscoe's speed.
• Step 2: Set up the formula for time, \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). This formula is crucial for comparing speeds over the distance.
• Step 3 & 4: Create time equations for Castroneves and Briscoe using the time formula. For Castroneves, it's \( \frac{2.5}{x + 0.694} \), and for Briscoe, it's \( \frac{2.492}{x} \).
• Step 5: Equalize the time equations to account for the fact that both drivers took the same amount of time.

Finally, solve the equation by cross-multiplying to isolate \( x \). Each of these steps guides us through logically figuring out unknown values by using relations described in the problem.

Unit Conversion in Speed Contexts

When dealing with racing speeds, it's essential to understand how to work with units correctly. The speeds are given in miles per hour (mph), which is a common unit in racing contexts. This exercise already uses converted units, so no additional conversion is necessary in the solution. However, knowing how conversion works is vital if the units differ.
The formula \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \) serves as a foundation for converting between units if needed. For instance, if speeds were given in kilometers per hour (km/h) instead, we would use conversion factors (1 mile ≈ 1.60934 kilometers) to switch these units to a consistent system. This consistency ensures the equations used to determine speeds hold true across different unit systems.