Problem 60 - Algebra A Combined Function

Question

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. A Hawaiian fruit company is studying the sales of a pineapple sauce to see if this product is to be continued. At the end of its first year, profits on this product amounted to \(\$ 30,000\). At the end of the fourth year, profits were \(\$ 66,000\). A Hawaiian fruit company is studying the sales of a pineapple sauce to see if this product is to be continued. At the end of its first year, profits on this product amounted to \(\$ 30,000\). At the end of the fourth year, profits were \(\$ 66,000\).

Step-by-Step Solution

Answer

The equation of the linear relationship is \( y = 12000x + 18000 \).

Step 1: Identify the Known Points

We are given two points on a linear graph: (1, 30000) and (4, 66000), where the first number in each pair represents the year and the second number represents the profit in dollars.

Step 2: Calculate the Slope

The slope \( m \) is calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the given points, we have \( m = \frac{66000 - 30000}{4 - 1} = \frac{36000}{3} = 12000 \).

Step 3: Write the Equation in Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + b \). We've found that \( m = 12000 \), and we can use one of the points to find \( b \). Using the point (1, 30000), we substitute into the equation: \( 30000 = 12000 \times 1 + b \). Solving for \( b \), we get \( b = 30000 - 12000 = 18000 \).

Step 4: Construct the Final Equation

Substituting back into the slope-intercept form equation, we have the complete equation: \( y = 12000x + 18000 \). This equation represents the relationship between the year and the profits.

Key Concepts

Understanding the Slope-Intercept Form

In linear equations, the slope-intercept form is a very popular way to write equations. It is represented as \( y = mx + b \). Here, \( y \) is the output value, \( m \) is the slope of the line, \( x \) is the input value, and \( b \) is the y-intercept. This format is quite straightforward:

• \( m \) tells us how steep or flat the line is.
• \( b \) illustrates where the line crosses the y-axis.

To use this form, like for the profit problem, you first need to find the slope which guides how profits change over time. Then, identify the y-intercept to know the starting point of the profits. This gives a complete picture of how variables are related linearly, making it easy to predict future values.

How to Calculate the Slope

The slope of a line indicates its steepness and direction. Calculating the slope involves comparing how a dependent variable (like profit) changes in response to the independent variable (like years). The formula for calculating the slope \( m \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here’s how you can do it:

• \( y_2 - y_1 \) is the change in the dependent variable.
• \( x_2 - x_1 \) is the change in the independent variable.

In the exercise, using points \((1, 30000)\) and \((4, 66000)\), we determined \( m = \frac{66000 - 30000}{4 - 1} = 12000 \). This means each year, the profits increase by \$12,000. The slope calculation tells us not just the direction, but also the rate of change.

Understanding Profit Growth Analysis with Linear Equations

Profit growth analysis using linear equations helps companies realize trends in their earnings over time. A linear equation in slope-intercept form simplifies the profit projection and analysis process. In the given example, the profits equation \( y = 12000x + 18000 \) illustrates how profits grow annually.

• The slope \( 12000 \) represents the annual profit increase, allowing for straightforward profit forecasting.
• The intercept \( 18000 \) indicates the starting profit value at the first data point, i.e., year one.

Analyzing this growth through linear equations, companies like the Hawaiian fruit company can decide strategic actions on whether to continue investing in products like pineapple sauce. The ability to interpret these mathematical concepts is crucial in making informed financial decisions.