Problem 61 - Algebra A Combined Function

Question

Solve. See Example 7. The Whammo Company has learned that by pricing a newly released Frisbee at \(\$ 6,\) sales will reach 2000 per day. Raising the price to \(\$ 8\) will cause the sales to fall to 1500 per day. Assume that the ratio of change in price to change in daily sales is constant, and let \(x\) be the price of the Frisbee and \(y\) be number of sales. a. Find the linear equation that models the price-sales relationship for this Frisbee. [Hint: The line must pass through (6,2000) and \((8,1500) .]\) b. Use this equation to predict the daily sales of Frisbees if the price is set at \(\$ 7.50\).

Step-by-Step Solution

Answer

The linear equation is \(y = -250x + 3500\). At $7.50, sales are 1625 Frisbees/day.

Step 1: Identify Given Points

We are given two points that represent the price and sales figures: (6, 2000) for a price of $6 yielding 2000 sales, and (8, 1500) for a price of $8 yielding 1500 sales. These points will be used to find the linear equation.

Step 2: Calculate the Slope

To find the slope (m) of the line, use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For our points, \((x_1, y_1)\) = (6, 2000) and \((x_2, y_2)\) = (8, 1500). Thus, the slope is \( m = \frac{1500 - 2000}{8 - 6} = \frac{-500}{2} = -250 \).

Step 3: Form the Linear Equation

Use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), with slope \(m = -250\) and one of the given points, let's use (6, 2000). Thus, the equation is: \[ y - 2000 = -250(x - 6) \] Simplify the equation: \[ y - 2000 = -250x + 1500 \] \[ y = -250x + 3500 \] This is the linear equation that models the price-sales relationship.

Step 4: Predict Sales for a Given Price

To predict the sales for a price of \(7.50, substitute \(x = 7.5\) into the linear equation \(y = -250x + 3500\). Thus, \[ y = -250(7.5) + 3500 \] Calculate the result: \[ y = -1875 + 3500 \] \[ y = 1625 \] This means if the price is set at \)7.50, the sales would be 1625 units per day.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a linear equation written as \( y = mx + b \). Here, \( m \) represents the slope of the line and \( b \) represents the y-intercept, which is where the line crosses the y-axis.
This form is useful because it quickly shows the rate of change (slope) and the starting point of the function (y-intercept).

When you have two points, like \((6, 2000)\) and \((8, 1500)\), you can find \( m \) by calculating the change in \( y \) over the change in \( x \), given as \( m = \frac{y_2 - y_1}{x_2 - x_1} \). With our points, this becomes \( -250 \), representing a decrease of 250 sales for every \( \$1 \) increase in price.
Finally, to find \( b \), substitute one point into the equation along with your slope and solve for \( b \). This gives you the full slope-intercept equation.

Price-Sales Relationship

The price-sales relationship shows how changes in the price of a product affect the quantity sold.
In our exercise, this connection is represented through a linear model. The model helps predict how changes in price influence demand, crucial for making pricing decisions.

The linear equation \( y = -250x + 3500 \) describes this relationship. It tells us that each increase of \( \\(1 \) will decrease sales by 250 units. The y-intercept of 3500 represents the sales when the Frisbee is offered at a theoretical price of \( \\)0 \), which is useful for understanding the baseline demand.
The negative slope means as price increases, sales decrease, a common trend in economics known as the law of demand.

• Negative Slope: Indicates inverse relationship.
• Intercept: Sales when price is zero.

Predicting Sales

Predicting sales involves substituting different price values into your linear equation to see the expected number of sales.
Here, we've used the equation \( y = -250x + 3500 \) to predict sales for a price of \( \\(7.50 \).

Substitute \( x = 7.5 \) into the equation to find:

• Calculate: \( y = -250(7.5) + 3500 \)
• Result: \( y = 1625 \)

This result tells us that at \( \\)7.50 \), approximately 1625 Frisbees would be sold per day.
Sales prediction based on price is a powerful tool for businesses to estimate revenue, manage inventory, and adjust marketing strategies based on expected demand.
By adjusting prices and using the linear model, businesses can fine-tune their approach to maximize both sales and profits.