Pack-Em-In has another development in the works. If it...

Pack-Em-In has another development in the works. If it builds 50 houses in this development, it will be able to sell them at $$\$ 190,000$$ each, but if it builds 70 houses, it will get only $$\$ 170,000$$ each. Obtain a linear demand equation and hence determine how many houses it should build to get the largest revenue. What is the largest possible revenue?

Key Concepts

Linear Demand Equation

This concept involves establishing a relationship between the price per item and the quantity sold. In the context of the problem, it represents how the sale price of a house changes linearly with the number of houses built. It generally takes the form p = a - bx, where p is the price per unit and x is the number of units, and the parameters a and b are determined by known conditions (such as given price-quantity pairs).

Revenue Function

The revenue function calculates the total income generated from sales, typically represented by the product of the number of units sold and the unit price. In this context, it is derived by combining the linear demand equation with the quantity, resulting in a quadratic function of the number of houses built. This function is used to find the production level that maximizes revenue.

Quadratic Optimization

This involves determining the maximum or minimum value of a quadratic function. Since the revenue function here is quadratic, maximizing revenue means finding the vertex of the parabola that represents this function. Quadratic optimization often involves techniques such as the vertex formula or completing the square.

Vertex of a Quadratic Function

The vertex of a quadratic function in standard form (ax² + bx + c) is the point where the function reaches its maximum or minimum value. For a downward-opening parabola (when a is negative), the vertex represents the maximum point. The x-coordinate of the vertex can be found using the formula x = -b/(2a), which in this problem gives the number of houses to build for maximum revenue.

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