Problem 38 - Algebra for College Students

Question

In \(2007,\) rain shortened the Indianapolis 500 race to \(415 \mathrm{mi}\). It was won by Dario Franchitti, who averaged \(151.774 \mathrm{mph}\). What was his time to the nearest thousandth?

Step-by-Step Solution

Answer

2.733 hours

Step 1: Identify Key Variables

Identify the distance and average speed provided. The distance is given as 415 miles and the average speed is 151.774 mph.

Step 2: Apply the Speed Formula

Use the formula for speed: \text{Speed} = \frac{\text{Distance}}{\text{Time}}. Rearrange the formula to solve for time: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).

Step 3: Calculate the Time

Substitute the given values into the formula: \( \text{Time} = \frac{415 \text{ miles}}{151.774 \text{ mph}} \).

Step 4: Simplify the Calculation

Perform the division to find the time: \( \text{Time} = 2.733 \text{ hours} \)

Step 5: Adjust to Required Precision

Round the result to the nearest thousandth: \( 2.733 \text{ hours} \).

Key Concepts

average speed calculations

Average speed is a measure of the distance traveled divided by the time it took to travel that distance. To find the average speed, you need two things: the total distance traveled and the total time taken. Here is how it's generally calculated:

If a car travels 300 miles in 5 hours, the average speed is calculated as follows:

• Average speed = Total distance / Total time
• Average speed = 300 miles / 5 hours = 60 mph

So, the car's average speed is 60 miles per hour. Even if the speed varies over the trip, this averages it out as if the car was traveling at a constant speed the entire time.

distance-time-speed formula

The formula for distance, time, and speed gives us a straightforward way to relate these three key variables. The important thing to remember is:

• Speed = Distance / Time

If you know any two of these variables, you can solve for the third. Let's break down the steps in our specific exercise:

• The distance traveled was 415 miles.
• The average speed was 151.774 miles per hour (mph).

To find the time it took to travel, we rearrange the formula to:

• Time = Distance / Speed

Substituting the values we have:

• Time = 415 miles / 151.774 mph ≈ 2.733 hours

This calculation shows that it took Dario Franchitti approximately 2.733 hours to complete the race.

precision and rounding in algebra

In algebra, precision and rounding are important for clear and accurate results, especially when dealing with measurements. Here's what you need to know:

• Precision refers to the detail in which a number is expressed.
• Rounding is reducing the digits in a number but keeping its value close to what it was.

In our exercise, when calculating time, we end up with a number containing several decimal places. To keep it simple and easier to report, we round it to the required precision—in this case, the nearest thousandth:

• The initial calculation gave us 2.733 hours.
• If the initial answer had been, say, 2.73314, and we needed to round to the nearest thousandth, it would still be 2.733 after rounding.

By rounding, we ensure the reported number is both accurate and manageable.