Problem 22 - Algebra 1: Concepts and Skills

Question

The fastest winning speed in the Daytona 500 is about 178 miles per hour. In the table below, calculate the distance traveled \(d\) (in miles) after time \(t\) (in hours) using the equation \(d=178 t\) For what values of \(t\) does the formula \(d=178 t\) correspond to the situation being modeled?

Step-by-Step Solution

Answer

The equation \(d = 178t\) will give the distance traveled when multiplied by the time in hours. It is applicable for any positive real values of \(t\), but in this context, \(t\) should represent a valid race duration.

Step 1: Identify the Given Variables

In this context, you are provided with the speed (178 miles per hour) and required to calculate distance for different time intervals using the formula \(d = 178t\).

Step 2: Apply the Formula

For a given time \(t\), multiply the known speed (178) by the time \(t\) to get the distance \(d\). So, if \(t = 1\) hour, the distance covered, \(d\) is \(178 \times 1 = 178\) miles. If \(t = 2\) hours, \(d = 178 \times 2 = 356\) miles. Continue this process for various values of \(t\).

Step 3: Discuss the Applicability of the Formula

The formula \(d = 178t\) is applicable for any positive real values of \(t\), as time can be any positive value. However, in the real-world context of a race, \(t\) would be the duration of the race which should be a reasonable figure. You would not consider unreasonably small (e.g., fractions of a second) or large values (e.g., days or weeks) for \(t\) in this context.

Key Concepts

Linear Equations

Linear equations are foundational in understanding many mathematical relationships. They are called 'linear' because when graphed on a coordinate plane, they form a straight line. In the equation \(d = 178t\), this is a linear equation where \(d\), the distance, is calculated as a function of time \(t\). The equation has the form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. In our equation, it's simpler – it takes the form \(d = speed \times time\), without a y-intercept because the line passes through the origin (0,0). This means as time increases, the distance traveled increases linearly. Understanding this relationship helps us quickly compute distance if we know the time and speed.

It is crucial to identify the variables in a problem involving linear equations:

• \(d\): The dependent variable (distance traveled)
• \(t\): The independent variable (time)
• 178: The constant representing speed

Linear equations ensure that the relationship remains consistent over different intervals, simplifying predictions.

Real-World Applications

The principle that connects speed, distance, and time is not restricted to merely textbook problems. It appears extensively in real-world applications. This simple equation, \(d = 178t\), can model real-life scenarios, particularly in transportation.

Consider a race car traveling at a constant speed of 178 miles per hour. This setup allows us to predict how far a car will travel over several hours or what time it will take to cover a certain distance.

• Travel Planning: Knowing the speed allows drivers to plan how long it will take to reach a destination.
• Logistics: Companies can estimate delivery times based on travel speeds and road conditions.
• Engineering: Understanding speed over time helps in designing more efficient transportation infrastructure.

The flexibility of linear equations makes them ideal for modeling predictable, uniform processes like travel under constant speed conditions.

Unit Rate

A unit rate is a comparison of any two separate but related measurements in which the second term is taken as one. In this context, the speed of 178 miles per hour is a unit rate – it reveals how many miles are traveled in one hour. Understanding and working with unit rates is crucial in interpreting the relationship between quantities like speed, time, and distance.

Unit rates are commonplace in many day-to-day scenarios. When you shop and compare prices, you often compare the cost per item or per pound – that's a unit rate. Similarly, when you check your car's fuel efficiency, you look at miles per gallon, which is also a unit rate.

Knowing the unit rate enables you to:

• Quickly compute distances for various hours by multiplying the time by 178.
• Estimate how changing conditions (like speed limits or traffic) can affect travel time.

By grasping the concept of unit rates, students and individuals can better manage calculations and make informed decisions in both academic and real-world settings.