Problem 65 - Algebra A Combined Function

Question

Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. The Pool Fun Company has learned that, by pricing a newly released Fun Noodle at \(\$ 3,\) sales will reach 10,000 Fun Noodles per day during the summer. Raising the price to \(\$ 5\) will cause sales to fall to 8000 Fun Noodles per day. a. Assume that the relationship between price and number of Fun Noodles sold is linear and write an equation describing this relationship. Use ordered pairs of the form (price, number sold). b. Predict the daily sales of Fun Noodles if the price is \(\$ 3.50\)

Step-by-Step Solution

Answer

a. The equation is \( y = -1000x + 13000 \). b. At \$3.50, sales are 9500 Fun Noodles.

Step 1: Define Ordered Pairs

We are given two points that describe the relationship between price and sales. These are (3, 10000) and (5, 8000), where the first value in each pair is the price and the second is the number sold.

Step 2: Calculate the Slope

The slope of the line is calculated as the change in the number sold divided by the change in price. The formula for slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting in our points, we get \( m = \frac{8000 - 10000}{5 - 3} = \frac{-2000}{2} = -1000 \).

Step 3: Write the Slope-Intercept Form

The equation of a line in slope-intercept form is \( y = mx + b \). We know the slope \( m = -1000 \). To find \( b \), substitute one of the points into the equation and solve for \( b \): \( 10000 = -1000 \times 3 + b \). Solving for \( b \) gives \( b = 13000 \). Therefore, the equation is \( y = -1000x + 13000 \).

Step 4: Predict Sales at a Given Price

To find the number of Fun Noodles sold at a price of \$3.50, substitute \( x = 3.50 \) into the equation \( y = -1000x + 13000 \): \( y = -1000 \times 3.50 + 13000 = -3500 + 13000 = 9500 \).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is essential for working with linear equations. A line's equation usually appears as \( y = mx + b \). Here, \( m \) stands for the slope, which indicates how steep the line is, and \( b \) represents the y-intercept, where the line crosses the y-axis.
This form is particularly valuable as it gives a straightforward way to identify both the slope and the y-intercept directly from the equation. So, when you come across a linear relationship, expressing it in this format will help you quickly understand and solve it.

• The slope \( m \) shows the rate of change, explaining how much \( y \) increases or decreases as \( x \) changes.
• The y-intercept \( b \) indicates the starting value of \( y \) when \( x = 0 \).

By putting this into practice, let's use the equation for the Fun Noodle sales. We've pinpointed the slope to be -1000, and the y-intercept to be 13000, thus forming the equation: \( y = -1000x + 13000 \).

Calculating Slope

Finding the slope of a line is like understanding the speed of a car. It tells you how fast the car is moving. In a linear equation, the slope \( m \) illustrates how quickly (or slowly) the dependent variable changes concerning the independent variable.
The formula for the slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula calculates the 'rise' (change in \( y \)) over the 'run' (change in \( x \)).
This process, using our original points \((3, 10000)\) and \((5, 8000)\), results in \( m = \frac{8000 - 10000}{5 - 3} = \frac{-2000}{2} = -1000 \).

• A negative slope indicates a downward trend; as price increases, sales decrease in this scenario.
• A positive slope would mean the opposite, with sales increasing as prices rise.

Knowing how to calculate the slope sheds light on the kind of relationship (positive or negative) that variables share.

Equations of Lines

Equations of lines summarize relationships between variables in a concise mathematical form. When dealing with a linear relationship, you're essentially looking at a problem where one variable depends on another in a consistent way.
Using the slope-intercept form, \( y = mx + b \), is particularly helpful for creating these equations because it neatly showcases both the slope and y-intercept.
For example, considering Fun Noodles' prices and sales, you've derived the equation \( y = -1000x + 13000 \). Here:

• \( y \) represents the number of Fun Noodles sold.
• \( x \) stands for the price.

This equation implies that for every one-dollar increase in price, sales decrease by 1000 units (indicated by the slope of -1000).
Furthermore, the y-intercept (13000) signifies the number of noodles that would be sold if they were free, emphasizing the importance of interpreting both \( m \) and \( b \) in understanding these relationships.