Problem 5 The distance in miles \(d\) that... [FREE SOLUTION]

Chapter 7: Problem 5

The distance in miles \(d\) that light travels in \(t\) seconds is given by the linear function \(d=186,000 t\). Find two solutions of this equation and describe what they mean.

Short Answer

Expert verified

In 1 second, light travels 186,000 miles; in 2 seconds, it travels 372,000 miles.

Step by step solution

Understand the function

The function given is a linear function: \( d = 186,000 t \). This equation shows the relationship between time \( t \) in seconds and the distance \( d \) in miles that light travels.

Choose a value for \( t \)

First, let's choose \( t = 1 \) second to calculate how far light travels. Substitute \( t = 1 \) into the function.

Calculate distance for \( t = 1 \)

Substitute \( t = 1 \) into \( d = 186,000 t \). Thus, \( d = 186,000 \times 1 = 186,000 \) miles. This means light travels 186,000 miles in 1 second.

Choose another value for \( t \)

Next, let's choose \( t = 2 \) seconds to find another distance.

Calculate distance for \( t = 2 \)

Substitute \( t = 2 \) into \( d = 186,000 t \). Thus, \( d = 186,000 \times 2 = 372,000 \) miles. This means light travels 372,000 miles in 2 seconds.

Interpret the solutions

The two solutions of the equation are: 1) \( d = 186,000 \) miles for \( t = 1 \) second, and 2) \( d = 372,000 \) miles for \( t = 2 \) seconds. Each solution represents how far light can travel in the given time.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distance-Rate-Time Relationships

Understanding the distance-rate-time relationship is essential when dealing with linear functions like the given equation, \( d = 186,000 t \). This equation is a straightforward representation of how distance, rate, and time are interrelated.

Rate: The constant 186,000 represents the speed of light in miles per second. It shows how fast light travels every second.
Time: This is represented by \( t \), the variable in the equation. It determines for how many seconds we are measuring the light's travel.
Distance: This is the result denoted by \( d \), which tells us how far the light has traveled in the given time \( t \).

The formula establishes that when you multiply time \( t \) by the rate of 186,000 miles per second, you get the total distance \( d \). It's a direct, linear relationship, meaning as time increases, distance increases proportionately.

Solving Equations

Solving equations involving linear functions like \( d = 186,000 t \) involves choosing values for one variable to find corresponding values of the other variable. Let's break it down:

Select a value for \( t \): Begin by choosing a time \( t \). For example, we chose \( t = 1 \) to see how far light travels in one second.
Substitute into the equation: Plug the selected \( t \) into the equation. For \( t = 1 \), the equation becomes \( d = 186,000 \times 1 \).
Calculate \( d \): Solve for \( d \) to find the distance. In our example, this gives \( d = 186,000 \) miles, showing that light travels 186,000 miles in one second.

Repeat this process with different values of \( t \) to see how the distance changes with time. This method of finding solutions gives us practical insights into how variables interact in a linear function.

Interpreting Solutions

Interpreting solutions in a linear equation involves understanding what the numbers and results mean in the context of the problem:

Solution: \( d = 186,000 \) for \( t = 1 \): This tells us that light travels 186,000 miles in one second of time. The number reflects the enormous speed at which light moves.
Solution: \( d = 372,000 \) for \( t = 2 \): In this case, light travels 372,000 miles over two seconds. The solution emphasizes the continuous and consistent rate of speed.

By interpreting these solutions, we gain insights into how distances accumulate over time at a constant rate. It showcases the predictive power of linear functions, allowing us to foresee future outcomes by setting precise conditions within the equation.

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91影视

The distance in miles \(d\) that light travels in \(t\) seconds is given by the linear function \(d=186,000 t\). Find two solutions of this equation and describe what they mean.

Short Answer

Step by step solution

Understand the function

Choose a value for \( t \)

Calculate distance for \( t = 1 \)

Choose another value for \( t \)

Calculate distance for \( t = 2 \)

Interpret the solutions

Key Concepts

Distance-Rate-Time Relationships

Solving Equations

Interpreting Solutions

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