Problem 96 - College Algebra and Calculus: An Applied Approach

Question

A line representing daily revenues \(y\) in terms of time \(x\) in days has a slope of \(m=100\). Interpret the change in daily revenues for a one-day increase in time.

Step-by-Step Solution

Answer

A one-day increase in time leads to an increase in daily revenues by \$100.

Step 1: Understand the Problem Context

The problem situation describes a line representing daily revenues in terms of time. The slope of the line \(m\) is given as 100, which in mathematical terms represents the rate of change of \(y\) with respect to \(x\). However, the problem asks for the interpretation of this slope in the context of daily revenues.

Step 2: Interpret the Slope in the Real-World Context

In the context of this problem, the slope refers to the rate of change of the daily revenues with respect to time. Since the slope is \(m = 100\), it means for every one-day increase in time, the daily revenue increases by 100 units. So, the interpretation of the slope in this context is that a one-day increase in time leads to an increase in daily revenues by 100 units.

Step 3: Formulate the Final Interpretation

Give the final interpretation of the slope in the problem's context: A one-day increase in time leads to an increase in daily revenues by \$100.

Key Concepts

Daily Revenue Increase

Understanding how daily revenue changes over time is critical for businesses to forecast earnings and manage budgets. In our exercise, a linear function is used to model the relationship between time (in days) and daily revenue. The line's slope, denoted as \( m = 100 \), is the number we'll focus on.

When we talk about a 'daily revenue increase', we're referring to how much more money a business makes on one day compared to the previous day. Interpreting the slope in this real-world scenario means that for each additional day, the revenue is expected to go up by the slope's value. So, if the slope is \( m = 100 \), it explicitly signifies that the business should anticipate earning an extra \$100 each day. This interpretation simplifies the rate of change in revenue into a daily figure that's easy to grasp and apply in a variety of business calculations.

Rate of Change

The 'rate of change' is a mathematical concept that tells us how one variable changes in relation to another. In a business context, deciphering the rate of change helps in understanding how quickly or slowly revenue is growing or declining over time.

In the case of our exercise, the rate of change is represented by the slope \( m \) of a linear function. A slope of \( m = 100 \) means there is a steady rate of increase in revenue; daily revenue goes up by \$100 for each additional day. Thus, the rate of change is constant: no matter how many days pass, the increase per day remains the same. Recognizing this rate of change provides a clear metric to gauge financial progress.

Linear Functions

Linear functions form the backbone of understanding relationships between two variables in algebra. They are depicted on a graph as straight lines and are defined by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

The slope \( m \) gives us the rate at which \( y \) changes for a one-unit increase in \( x \). For our daily revenue problem, the linear function tells us that the revenue's increase is predictable and consistent over time. This makes linear functions extremely valuable for projecting future events based on past data. In businesses, they can serve as a simplified model to predict outcomes, like revenue or costs, as long as the growth remains steady and linear.