Problem 1 - Algebra Form and Function

Question

A homing pigeon starts 1000 miles from home and flies 50 miles toward home each day. Express distance from home in miles, \(D,\) as a function of number of days, \(d\).

Step-by-Step Solution

Answer

Question: A homing pigeon starts 1000 miles away from home and flies 50 miles toward home each day. Express the distance from home, D, as a function of the number of days, d. Answer: D(d) = 1000 - 50d

Step 1: Identify the initial distance, rate, and variable

The problem states that the initial distance is 1000 miles and the homing pigeon flies 50 miles toward home each day. Thus, the initial distance is 1000 miles, the rate is 50 miles per day, and the variable we want to express is the distance from home, D, as a function of the number of days, d.

Step 2: Write the equation for distance

The equation to express distance as a function of time is: D = initial distance - rate * time Since we know the initial distance is 1000 miles and the rate is 50 miles per day, we can plug these values into the equation: D = 1000 - 50d

Step 3: Write the function D(d)

Now that we have the equation for D in terms of d, we can write the function D(d) as follows: D(d) = 1000 - 50d This function represents the distance from home, D, as a function of the number of days, d.

Key Concepts

Distance-Time Relationship

The distance-time relationship is a way to express how distance changes over time. It's a concept frequently used in physics and mathematics to understand movement. In this particular exercise with the homing pigeon, we see how its journey home can be modeled as a function of time.

Here's how it works: the pigeon starts at a specific distance from home, which is 1000 miles in this case. Each day, it covers 50 miles, decreasing the distance remaining. Thus, over time, the total distance from home reduces, adhering to a fixed rate of travel.

This simple linear relationship helps us to predict the pigeon’s position over a continuous timeline. By knowing the number of days the pigeon has been flying, you can calculate exactly how far it is from home at any given point.

Slope-Intercept Form

The slope-intercept form of a linear equation is essential for expressing relationships such as the one described in our pigeon problem. This form is typically written as \( y = mx + b \), where \( m \) represents the slope, and \( b \) is the y-intercept.

In the case of the pigeon exercise, the distance function \( D(d) = 1000 - 50d \) is already in slope-intercept form. The equation reveals that the distance \( D \) decreases by 50 miles each day, which is represented by the slope \(-50\). The y-intercept, \( 1000 \), indicates the initial distance from home.

Understanding the slope, negative here, informs us that as time progresses (i.e., the days increase), the distance to home decreases—this is characteristic of linear functions involving a diminishing quantity over a set period.

Function Notation

Function notation is a way to describe functions in mathematics—it's a concise method to show relationships between variables. Here, the distance from home is denoted as \( D(d) \), where \( D \) is the function and \( d \) represents the number of days.

Using \( D(d) \) allows us to easily understand and compute the distance based on any given day. You plug in a value for \( d \) (number of days), and the function tells you the remaining distance, \( D \). It simplifies complex expressions, offering a clearer view of how variables relate.

Function notation is more than mathematical convenience; it's a tool that highlights how input and output are interconnected in a formulaic context. In the pigeon scenario, it becomes incredibly helpful in tracking the changes in distance from home over time.