Problem 52 - Algebra Form and Function

Question

A company's profit after \(t\) months of operation is given by \(P(t)=1000+500(t-4)\) (a) What is the practical meaning of the constants 4 and \(1000 ?\) (b) Rewrite the function in slope-intercept form and give a practical interpretation of the constants.

Step-by-Step Solution

Answer

Additionally, what do the slope and y-intercept represent when the function is rewritten in slope-intercept form? Answer: In the profit function P(t) = 1000 + 500(t - 4), the constant 4 represents the time (in months) it takes for the company to reach a profit of 1000 dollars. The constant 1000 represents the initial profit that the company reaches after 4 months of operation. When rewritten in slope-intercept form as P(t) = 500t - 1000, the slope of 500 represents the rate at which the company's profit increases by 500 dollars per month after reaching the initial 1000 dollars profit in the 4th month. The y-intercept of -1000 represents the starting profit, which is a negative value that turns positive by the 4th month.

Step 1: Understand the given profit function

The given profit function is P(t) = 1000 + 500(t - 4). This function relates the company's profit P(t) to the number of months t the company has been in operation.

Step 2: Analyze the constants in the profit function

In the given profit function, let's first look at the constant 4. This number is being subtracted from the time variable t. This means that when t = 4, the term (t - 4) becomes zero, and the profit function becomes P(4) = 1000. Hence, 4 represents the time (in months) it takes for the company to reach a profit of 1000 dollars. Next, the constant 1000 is the value of the profit function when t = 4. It indicates the initial profit (in dollars) that the company reaches after 4 months of operation.

Step 3: Rewrite the function in slope-intercept form

The slope-intercept form of a linear function is given by y = mx + b, where m is the slope and b is the y-intercept. In the given profit function P(t) = 1000 + 500(t - 4), we can rewrite it in the slope-intercept form by expanding the product term and simplifying: P(t) = 1000 + 500(t - 4) P(t) = 1000 + 500t - 2000 P(t) = 500t - 1000 Now, the function is in the form P(t) = mt + b, with m = 500 and b = -1000.

Step 4: Analyze the constants in the slope-intercept form

In the slope-intercept form P(t) = 500t - 1000, the slope m = 500 represents the rate at which the company's profit increases with respect to time (in months). In other words, the company's profit increases by 500 dollars per month after they reach the initial 1000 dollars profit in the 4th month. The constant b = -1000 is the y-intercept, which is the profit value when t = 0 (i.e., the starting profit). Since the company starts with a negative profit that will be cancelled out and turned positive by the 4th month, it makes sense for the y-intercept to be a negative value.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing the equation of a straight line. It is written as \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept. This format is incredibly useful because it allows you to quickly understand how the line behaves.

• The slope \(m\) tells you how much the y-value (profit, in this case) changes for every one unit increase in the x-value (months).
• The y-intercept \(b\) tells you the starting value of y (the profit) when the x-value is zero.

For the profit function \(P(t) = 500t - 1000\),

• The slope \(m = 500\) indicates that when the company operates for one more month, the profit increases by \(500.
• The y-intercept \(b = -1000\) means that before the company starts its operations, it has a balance of -\)1000, a hypothetical point that explains the initial financial situation.

Constants in Algebra

In algebra, constants are numbers that have a fixed value. They do not change with different variables. In the context of the profit function \(P(t) = 1000 + 500(t - 4)\), we encounter two specific constants, 4 and 1000.

• The number 4 is part of the expression \(t - 4)\). It represents a specific point in time—exactly 4 months into the company's operations. When \(t = 4\), this term becomes zero, indicating a milestone in profit calculations.
• The constant 1000 is added to the product of 500 and \((t - 4)\). It signifies the profit value the company reaches at 4 months, showing a benchmark of financial success.

Understanding these constants is essential for interpreting the behavior of the equation. They help paint a clearer picture of how the profit evolves over time, linking algebraic expressions to real-world meanings.

Profit Function

A profit function models how a company's profits change over time, using mathematical expressions to translate real-world operations into understandable data. In our example, the profit function is given by \(P(t) = 1000 + 500(t - 4)\). This tells you about the company's profit in relation to time \(t\).

• The function includes a starting term, \(1000\), showing a profit value at 4 months that effectively begins your observation of the company's performance.
• The term \(500(t - 4)\) reveals how profit increases each month after hitting that 4-month threshold at a rate of $500 per month.

The function was later rewritten to the slope-intercept form \(P(t) = 500t - 1000\), providing a more straightforward view.This transformation maintains accuracy while clarifying relationships, making analysis and understanding much easier. The rate of change (the slope here) and initial offset (the intercept) in the slope-intercept form are vital components that simplify how we communicate about growing profits over months.