Problem 23 - Algebra Form and Function

Question

Identify the initial value and the rate of change, and explain their meanings in practical terms. The distance, \(d\), in meters from the shore, of a surfer riding a wave is given by \(d=120-5 t,\) where \(t\) is the number of seconds since she caught the wave.

Step-by-Step Solution

Answer

Answer: The initial distance of the surfer from the shore is 120 meters, which means she started 120 meters away from the shore when she caught the wave. The rate of change of her distance with respect to time is -5 meters per second, meaning she is moving towards the shore at a rate of 5 meters per second while riding the wave.

Step 1: Identifying the Initial Value

To find the initial value, we need to determine the distance, \(d\), when \(t = 0\). Plug in \(t = 0\) into the given equation: \(d = 120 - 5(0)\) \(d = 120\) So the initial value is 120 meters. In practical terms, this means that when the surfer first caught the wave, she was 120 meters away from the shore.

Step 2: Identifying the Rate of Change

The given equation is \(d = 120 - 5t\). The rate of change is the coefficient of the variable \(t\), which is \(-5\). This means that for every second since she caught the wave, her distance from the shore is decreasing at a rate of 5 meters per second. In practical terms, this means that the surfer is moving towards the shore at a rate of 5 meters per second while riding the wave.

Key Concepts

Initial Value

In the context of a linear function, the initial value represents the starting point of a situation, usually when the time or the independent variable is zero. In mathematical terms, it is the y-intercept of the equation. Consider the equation given: \(d = 120 - 5t\). Here, the initial value can be identified by setting \(t = 0\). This will give us:\[d = 120 - 5(0) = 120\]This tells us that at \(t = 0\), the distance \(d\) is 120 meters. Thus, the initial value is 120 meters. In practical terms, this initial value represents the surfer's distance from the shore at the exact moment when she catches the wave. It's important as it sets the scene for her subsequent movement towards the shore.

Rate of Change

The rate of change in a linear function indicates how quickly the dependent variable changes with respect to the independent variable. It's represented by the slope of the linear equation. In this case, our equation is \(d = 120 - 5t\).- The coefficient of \(t\), which is \(-5\), represents the rate of change.- A rate of \(-5\) means that for each second counted by \(t\), the distance \(d\) decreases by 5 meters.This negative rate of change is what drives the surfer towards the shore. If you think about it physically, with each passing second, the surfer gets 5 meters closer to the shore. Understanding this helps us see how quickly she progresses during the wave ride.

Practical Interpretation

The practical interpretation of a linear equation like \(d = 120 - 5t\) involves understanding what the initial value and the rate of change mean in a real-world scenario. Here, we examine the surfing context:

• The initial value (120 meters) shows the surfer's starting position in relation to the shore when she first begins riding the wave.
• The rate of change, \(-5\) meters per second, indicates how the surfer approaches the shore. It tells us she's moving towards land at a consistent pace of 5 meters every second.

This equation as a whole gives us a complete picture of the surfer's movement. It starts with where she began on the wave and then describes how quickly she moves toward the shore. This seamless merging of mathematics and practical life underscores the beauty of linear functions. These concepts together show not just a line on a graph but real-life dynamics that can be observed and appreciated.