Problem 13 - Algebra Form and Function

Question

In Exercises \(13-23,\) identify the initial value and the rate of change, and explain their meanings in practical terms. An orbiting spaceship releases a probe that travels directly away from Earth. The probe's distance \(s\) (in km) from Earth after \(t\) seconds is given by \(s=600+5 t\).

Step-by-Step Solution

Answer

In this problem, we are given the linear function \(s = 600 + 5t\) that represents the distance (s) of a probe from Earth in kilometers (km) at any given time (t) in seconds (s). The initial value is 600 km, meaning that the probe starts 600 km away from Earth. The rate of change is 5 km/s, indicating that the probe is moving away from Earth at a constant speed of 5 km every second.

Step 1: Identifying the initial value

To find the initial value, we need to evaluate the function at \(t = 0\). Plugging in \(t = 0\) into the given equation \(s = 600 + 5t\), we get: \(s = 600 + 5(0) = 600\). So, the initial value is 600 km.

Step 2: Identify the rate of change

The rate of change for a linear function in slope-intercept form can be found by looking at the coefficient of \(t\). In this case, the rate of change is 5 km/s.

Step 3: Explain the initial value and rate of change in practical terms

The initial value of 600 km represents the distance between the probe and Earth when the spaceship first releases the probe (\(t = 0\)). In other words, the probe starts 600 km away from Earth. The rate of change of 5 km/s represents the speed at which the probe is moving away from Earth. This means the probe's distance from Earth increases by 5 km every second.

Key Concepts

Initial Value

In a linear function, the initial value is the point where the function begins. It's like the starting point in a race. Imagine if you are running a marathon; your initial position is the line where you first begin.
In mathematical terms, the initial value is what you get when you set the independent variable (usually time, represented by \( t \)) to zero.
In our exercise, the linear function is given as \( s = 600 + 5t \). When we set \( t = 0 \), the equation becomes \( s = 600 + 5(0) = 600 \).
This tells us that the initial value is 600 km, signifying that the probe starts its journey 600 km away from Earth, right when it is released by the spaceship.
Understanding the initial value helps us grasp the starting position or condition of any scenario depicted by a linear equation. Whenever you see a linear function, try setting \( t \) or another variable to zero to find this key piece of information.

Rate of Change

The rate of change in a linear function determines how quickly or slowly the dependent variable changes with respect to the independent variable. Think of it as the speed of a moving car; it's how fast or slow the car is going.
For our example \( s = 600 + 5t \), we find the rate of change by looking at the coefficient of \( t \), which is 5.
This means that with every passing second, the distance of the probe from Earth increases by 5 km.
The rate of change is a crucial concept, as it tells us how the situation develops over time or another variable.

• In economics, it could represent how costs rise with every product made.
• In physics, it might denote how a speed changes over time.

In simplest terms, the rate of change tells us everything about the 'speed' of change in the linear relationship being looked at. Without this, we wouldn't know how our initial state evolves.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of writing the equation so it's easy to read both the initial value (or intercept) and the rate of change (or slope). It is typically written as \( y = mx + b \), where:
\( m \) represents the slope or rate of change, and
\( b \) is the y-intercept or initial value.
For our specific situation with the probe, the equation is \( s = 600 + 5t \), already in slope-intercept form. We can tell right away that:

• \( b = 600 \), meaning the initial distance from Earth is 600 km.
• \( m = 5 \), indicating the probe moves away from Earth at 5 km each second.

Using the slope-intercept form makes it easier to plug in values, predict outcomes, and understand the dynamics of linear relationships at a glance. Whether you're dealing with simple physics problems or complex data graphs, this form helps you cut through the confusion and see the fundamental bits: where you start and how fast you’re heading towards a goal.