Problem 20 - 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 \(d=178 t\) Copy and complete the input-output table. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Time (hours) } & {0.25} & {0.50} & {0.75} & {1.00} & {1.25} & {1.50} \\ \hline \text { Distance traveled (miles) } & {?} & {?} & {?} & {?} & {?} & {?} \\\ \hline \end{array} $$

Step-by-Step Solution

Answer

After substituting the given times in hours (0.25, 0.50, 0.75, 1.00, 1.25, and 1.50) into the formula \(d = 178t\), you will get the distances traveled as follows 44.5 miles, 89 miles, 133.5 miles, 178 miles, 222.5 miles, and 267 miles respectively.

Step 1: Understand the Problem

In this problem, a racing car's speed is given to be 178 miles per hour. This constant speed relationship between distance and time is described by the formula \(d = 178t\), where \(d\) represents the distance traveled in miles and \(t\) represents the time in hours.

Step 2: Calculate the Distance for Each Time

Substitute each given time into the formula and solve to find the corresponding distance. The formula becomes:For \(t = 0.25\) hours, \(d = 178(0.25)\)For \(t = 0.50\) hours, \(d = 178(0.50)\)For \(t = 0.75\) hours, \(d = 178(0.75)\)For \(t = 1.00\) hours, \(d = 178(1.00)\)For \(t = 1.25\) hours, \(d = 178(1.25)\)For \(t = 1.50\) hours, \(d = 178(1.50)\

Step 3: Complete the table

Once the distances have been calculated, fill in the ‘Distance traveled (miles)’ row in the table as follows: \[\begin{array}{|l|c|c|c|c|c|} \hline \text { Time (hours) } & {0.25} & {0.50} & {0.75} & {1.00} & {1.25} & {1.50} \ \hline \text { Distance traveled (miles) } & {44.5} & {89} & {133.5} & {178} & {222.5} & {267} \\hline \end{array} \]

Step 4: Review the Solution

Review the solution to ensure its correctness by making sure all times reported are calculated correctly and all computations are executed properly. Always keep the mathematical relationship between time and distance in mind, remembering that longer times should yield longer distances.

Key Concepts

Distance-Speed-Time Relationship

The relationship between distance, speed, and time is a fundamental concept in physics and mathematics. This concept helps us understand how far an object travels over a certain period of time at a given speed.

• Distance is how far an object moves. It is usually measured in miles or kilometers.
• Speed is how fast an object moves. It's the rate at which the distance is covered, typically measured in miles per hour (mph) or kilometers per hour (kph).
• Time refers to how long the object is moving, often measured in seconds, minutes, or hours.

The formula to calculate the distance traveled by an object when you know the speed and time is: \[ d = ext{{speed}} imes ext{{time}} \] In the Daytona 500 exercise, the given speed is 178 mph, and the time varies. Therefore, the distance traveled can be calculated using the equation \( d = 178t \).
This equation can be used to find the distance traveled at any given time by simply substituting the time in hours into the formula.

Constant Speed

The concept of constant speed is essential for solving linear distance problems like the one presented. A constant speed means that the object travels the same distance every hour.
When you are traveling at a constant speed:

• The speedometer stays at the same reading.
• The object covers equal distances in equal intervals of time.

For the Daytona 500 scenario, the car is traveling at a constant speed of 178 miles per hour. This allows us to predict the distance traveled at any time simply by using multiplication.
For instance, if the car travels for 1 hour, it will cover 178 miles. Similarly, if it travels for 1.5 hours, the distance covered can be calculated as \( d = 178 imes 1.5 = 267 \) miles.
Constant speed simplifies calculations because it creates a linear relationship between time and distance.

Table of Values

A table of values is a useful tool to organize and represent the relationship between two quantities, such as time and distance, especially when they depend on each other.
For problems involving constant speed, you can use a table of values to see how distance changes as time progresses. This is particularly helpful for visual learners who benefit from seeing numerical patterns.

• Each column in the table corresponds to a specific point in time.
• The rows underneath each time show the calculated distances based on the equation \( d = 178t \).

In this problem, by substituting different values of time (like 0.25 hours, 0.50 hours, etc.) into our equation \( d = 178t \), we get the distances: 44.5 miles, 89 miles, and so on.
Creating a table helps break down calculations into manageable steps, making it easier to verify results and understand the linear relationship.