Solve. The Whammo Company has learned that by pricing a...

Solve. The Whammo Company has learned that by pricing a newly released flying disc 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 flying disc and $y$ be number of sales. a. Find the linear equation that models the price-sales relationship for this flying disc. [Hint: The line must pass through $(6,2000)$ and $(8,1500)$.] b. Use this equation to predict the daily sales of flying discs if the price is set at $$\$ 7.50$$.

Solve.
The Whammo Company has learned that by pricing a newly released flying disc 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 flying disc and $y$ be number of sales.
a. Find the linear equation that models the price-sales relationship for this flying disc. [Hint: The line must pass through $(6,2000)$ and $(8,1500)$.]
b. Use this equation to predict the daily sales of flying discs if the price is set at $$\$ 7.50$$.

Key Concepts

Slope

The slope of a line represents the constant rate of change between the dependent and independent variables. It quantifies how much one variable changes, on average, for a unit change in the other variable. The slope is crucial for understanding the direction and steepness of the line in any linear model.

Prediction in Linear Models

Using linear models for prediction involves substituting a specific value of the independent variable into the linear equation to estimate the corresponding value of the dependent variable. This concept is widely used in various fields to make forecasts based on past observed trends.

Intercept

The intercept in a linear equation is the value at which the line crosses the vertical axis. This represents the value of the dependent variable when the independent variable is zero, offering insight into the baseline level of the relationship modeled.

Linear Relationship

A linear relationship describes a straight-line connection between two variables, where the change in one variable is constant with respect to the change in the other. This concept is fundamental in modeling scenarios where changes occur at a constant rate and can be represented by a linear equation.

Point-Slope Form

The point-slope form is a method of writing the equation of a line when you know its slope and one point on the line. This form is particularly useful in constructing a linear model because it directly incorporates key known values from which the overall linear relationship can be derived.

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