Question

It takes 4.25 years for starlight to travel 25 trillion miles. Let \(t\) be the number of years and let \(f(t)\) be trillions of miles traveled. Write a linear function \(f(t)\) that expresses the distance traveled as a function of time.

Short answer

The linear function that expresses distance traveled as a function of time is \(f(t) = 5.88t\).

Step-by-step solution

Identify given values

We are given that in 4.25 years, 25 trillion miles are covered. Let's call the number of years 't', and the distance in trillions of miles 'f(t)'.

Determine the rate of change (slope)

The rate of change of a linear function is determined by calculating the change in the output value (f(t)) divided by the change in the input value (t). Here, that means calculating the number of trillions of miles traveled per year. So we divide 25 trillion miles by 4.25 years. This gives us a rate of \((25/4.25) ≈ 5.88\) trillions of miles per year.

Construct the linear function

Now that we have determined the slope, we can create our linear function. The form of a linear function is \(f(t) = mt + b\), where m is the slope and b is the y-intercept. Given that at time 0, the distance travelled is 0, our y-intercept (b) will be 0. Substituting our values into this equation, our function becomes \(f(t) = 5.88t\).

Key concepts

Understanding Rate of Change

The rate of change in algebra is a measure of how one quantity changes in relation to another. It is a concept that is fundamental in understanding linear functions, particularly when we model real-world situations. When we talk about the rate of change, we usually refer to the slope of a line in a linear equation. For instance, in the context of the exercise, the rate of change represents how far light travels in one year, or the speed of light.

In simple terms, if you think of a car traveling at a constant speed, the rate of change would be this constant speed — it tells you how much distance the car covers each hour.

To find the rate of change in the given exercise, one must divide the change in distance (25 trillion miles) by the change in time (4.25 years), which results in approximately 5.88 trillion miles per year. This number, 5.88, serves as the slope (m) in our linear function and helps to understand how quickly the starlight is covering distance over time.

• Rate of change is often denoted as the slope in a linear function.
• It is calculated as the change in the output value divided by the change in the input value.
• In the context of distance over time, it represents speed or velocity.

Writing Linear Equations

Writing linear equations is a critical skill in algebra. It involves expressing relationships between two variables in the form of a line. This is typically done using the slope-intercept form of a line, which is represented as:
\[ y = mx + b \] where 'm' is the slope and 'b' is the y-intercept. The slope signifies the rate of change as just discussed, and the y-intercept represents the value of 'y' when 'x' is zero.

In our exercise, writing a linear equation allows us to model the relationship between time and distance for the starlight. With a slope of 5.88, which indicates the number of trillions of miles light travels per year, and knowing that at time zero the distance is also zero, we can deduce that the y-intercept (b) is 0. This results in the linear function: \[ f(t) = 5.88t \] This linear equation is a powerful tool allowing us to calculate the distance traveled by light for any given time in years.

• Linear equations can model relationships between two variables.
• The slope-intercept form is the most common way to write these equations.
• The y-intercept is the starting value of the dependent variable when the independent variable is zero.

Function of Time

A function of time is an expression that describes how a certain quantity changes as time progresses. It's a fundamental concept in both mathematics and physics, as many processes depend on time. In algebra, a linear function of time will have the form \[ f(t) = mt + b \], where 't' represents time.

In the scenario presented in the exercise, the quantity we are examining is the distance that starlight travels, which is a function of time. By establishing this function, we can see the direct proportionality between time and distance traveled, since we're dealing with a linear relationship.

The linear function \[ f(t) = 5.88t \] allows anyone to plug in a value for 't', the number of years, and immediately determine the corresponding distance in trillions of miles that the starlight will have traveled. It gives a clear and direct way to see the impact of time on the journey of starlight through space.

• A function of time relates the change in a quantity to the passage of time.
• In linear functions, this relationship is proportionate and consistent.
• The function \[ f(t) = mt + b \] encapsulates this concept with 'm' as the rate at which the quantity changes over time.