Problem 22 - College Algebra

Question

For the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year. If the function \(W\) is graphed, find and interpret the slope of the function.

Step-by-Step Solution

Answer

The slope is 0.5, indicating a monthly weight gain of 0.5 pounds.

Step 1: Understand the problem

The question is asking for the slope of the function that represents the weight of the baby over time. We are given the initial weight and the rate of weight gain over a period.

Step 2: Define the variables and formula

Let the variable \(x\) represent the number of months since birth, and \(W(x)\) represent the baby's weight. We know the baby starts at 7.5 pounds and gains 0.5 pounds per month. The function can be represented as \(W(x) = 7.5 + 0.5x\).

Step 3: Identify the slope

The slope of a linear function in the form \(W(x) = mx + b\) is \(m\). In our function \(W(x) = 7.5 + 0.5x\), the slope \(m\) is 0.5.

Step 4: Interpret the slope

The slope of 0.5 indicates that the baby's weight increases by 0.5 pounds each month. This constant rate of change reflects the weight gain over time.

Key Concepts

Slope Interpretation

The slope of a linear function is an essential concept in understanding how changes occur over time. In simple terms, the slope describes how steep a line is. It's the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.
In mathematical terms, the slope is often represented by the letter "m" in the linear equation format, written as: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. For our specific problem, the graphing function is \( W(x) = 7.5 + 0.5x \). Here, the slope \( m \) is 0.5.

• For each month (which is our horizontal change), the weight increases by 0.5 pounds (our vertical change).
• Thus, every additional month consistently adds 0.5 pounds to the baby's weight, represented visually by the upward slant of the graph.

Rate of Change

The rate of change in a linear function is a way to describe how a quantity changes in relation to another. In the context of our problem, it refers to how the baby's weight changes as time progresses. The key feature of linear functions is that they have a constant rate of change.
This means that the weight gain per month does not vary; it remains the same throughout the period we are examining. In our scenario:

• The rate of change is 0.5 pounds per month.
• This consistency shows a steady progression, making predictions about future weight straightforward.

The idea of a constant rate makes linear functions especially useful in modeling behaviors that grow or decline uniformly over time. Predicting future values becomes simpler because you know the rate remains unchanged.

Weight Gain Modeling

Modeling weight gain using a linear function provides a clear picture of how a baby's weight changes over time. By using the function \( W(x) = 7.5 + 0.5x \), where \( W(x) \) represents the weight in pounds and \( x \) is the number of months, we gain insights into weight patterns.
1. **Starting Point**: The starting weight of 7.5 pounds is our baseline—all future weights are calculated based on this starting point.2. **Consistent Increase**: The function clearly forecasts a monthly weight gain of 0.5 pounds. Since the slope is constant, we can determine future weights with ease.
For example, at 6 months:

• You plug \( x = 6 \) into the function: \( W(6) = 7.5 + 0.5(6) = 10.5 \) pounds.
• Similarly, at 12 months: \( W(12) = 7.5 + 0.5(12) = 13.5 \) pounds.

This straightforward calculation allows us to model not just weight, but any scenario where a steady increase or decrease is observed over time.