By setting up a linear-quadratic system and solving it graphically, we can determine the price at which the business will earn $50,000. The solution set represents the specific selling price that will result in the desired profit.
a. To create a linear-quadratic system to determine the price for which the business will earn $50,000, we need to set up an equation by equating the profit function to $50,000.
The profit function is given as P = 2x² + 30x, where x represents the selling price of the items.
Setting P = 50,000, the equation becomes:
2x² + 30x = 50,000
b. To solve the system graphically, we can plot the graph of the quadratic function P = 2x² + 30x and the line y = 50,000 on the same coordinate plane. The x-coordinate where the quadratic function intersects the line represents the price for which the business will earn $50,000.
By graphing the two equations, we can find the point of intersection, which gives us the price at which the profit is $50,000.
c. The solution set of the system represents the price at which the business will earn $50,000. In other words, it is the selling price of the items that will result in a profit of $50,000. The solution provides a specific value for x, which can be used to determine the price the business should set for the items to achieve the desired profit.