Problem 30 - Algebra Form and Function

Question

Does the description lead to a linear function? If so, give a formula for the function. The distance traveled is the speed, \(45 \mathrm{mph}\), times the number of hours, \(t\).

Step-by-Step Solution

Answer

If so, provide a formula for the function. Answer: Yes, the description leads to a linear function. The formula for the function is \(d(t) = 45t\), where \(d\) represents the distance traveled and \(t\) represents the time in hours.

Step 1: Determine the equation

The distance traveled is the product of the speed and the time, which can be represented as \(d = 45t\). In this equation, \(d\) represents the distance, \(t\) represents the time in hours, and 45 is the given speed in miles per hour.

Step 2: Identify the type of function

The equation \(d = 45t\) is a linear equation because the power of the variable (i.e., \(t\)) is 1 and there is no other term containing a variable. Therefore, the description leads to a linear function.

Step 3: Formula for the linear function

Since the equation is linear, the formula for the function can be written in slope-intercept form: \(d(t) = mt + b\). For this problem, the slope (\(m\)) is 45 (the coefficient of \(t\)) and the y-intercept (\(b\)) is 0 (as there is no constant term). The formula for the linear function representing the distance traveled as a function of time is: \(d(t) = 45t\)

Key Concepts

distance formula

The distance formula helps us calculate the distance traveled over time when speed is constant. In the context of linear functions, we often think about distance in terms of a straight line, connecting different points. If you know two points on this line, you can easily calculate the distance between them.

In our problem, the formula can be seen as a relationship between speed and time. If you drive at a constant speed of 45 mph, then the distance traveled can be found by multiplying the speed by the number of hours you've been traveling.

• Distance, \(d\), equals speed times time: \(d = 45t\)
• This is a straightforward linear relationship because of the constant speed.

Understanding this simple calculation demonstrates how the distance formula, although basic, lays the groundwork for solving real-world problems involving distance and travel.

slope-intercept form

The slope-intercept form is a way of writing algebraic equations so they're easy to understand and interpret.

In any linear equation, the slope-intercept form is expressed as \(y = mx + b\), where:

• \(m\) is the slope of the line.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

In our exercise, the linear function of distance is written as \(d(t) = 45t\). By comparing it to the general slope-intercept form, we identify that:

• The slope \(m\) is 45, which represents the speed (45 mph).
• The y-intercept \(b\) is 0, meaning the line passes through the origin. This makes sense because if no time has passed, no distance has been traveled.

This form makes it easy to graph the function and quickly understand the relationship between distance, speed, and time.

equation of a line

An equation of a line in mathematics is a statement that describes a straight line, typically using a relationship between independent and dependent variables.

In the context of our problem, the equation \(d = 45t\) describes how distance changes with respect to time. Every straight line equation will have a slope, which defines its inclination, and an intercept, which tells where the line crosses a particular axis.

• **Slope**: This measures the steepness of the line. In this case, each hour adds 45 miles to the distance due to the slope of 45.
• **Intercept**: Corresponds to the starting point of the line. Here, the intercept is zero, indicating that at time zero, the distance is also zero.

We use the slope to discern how much and in what direction the line from the equation tilts. Understanding this can help predict and make decisions based on linear data trends, such as estimating future travel distances given a steady speed.