During the summer months Terry makes and sells necklaces on...

During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $$ 10$ each and his sales averaged 20 per day. When he increased the price by S1, he found that the average decreased by two sales per day. (a) Find the demand function, assuming that it is linear. (b) If the material for each necklace costs Terry $$ 6,$ what should the selling price be to maximize his profit?

During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $$ 10$ each and his sales averaged 20 per day. When he increased the price by
S1, he found that the average decreased by two sales per day.
(a) Find the demand function, assuming that it is linear.
(b) If the material for each necklace costs Terry $$ 6,$ what should the selling price be to maximize his profit?

Key Concepts

Vertex of a Quadratic Function

For a quadratic function representing profit, where the coefficient of the squared term is negative (indicating a concave downward curve), the maximum profit occurs at the vertex. The vertex can be computed using the formula -b/(2a), where a and b are coefficients from the quadratic expression, which simplifies the process of finding the optimal price or quantity.

Profit Maximization

Profit maximization involves finding the price or quantity that yields the highest profit. It often involves forming a quadratic profit function based on revenue and cost, and then using methods such as taking the derivative or applying the vertex formula to determine the optimal conditions under which profit is maximized.

Demand Function

The demand function represents the relationship between the price of a product and the quantity that consumers are willing to purchase. In this context, it is assumed to be linear, meaning it can be expressed in the form Q = a - bP, where changes in price have a constant impact on the quantity sold.

Linear Model

A linear model assumes that the relationship between variables is a straight line. This simplifies analysis by allowing us to use a simple equation with constant parameters, such as the slope and intercept, making it easier to predict how changes in one variable affect another.

Revenue and Cost Functions

Revenue is calculated as the product of the selling price and the quantity sold, while the cost function represents the total cost incurred, which can be dependent on the quantity produced or sold. In problems like this, where a per-unit cost is provided, the profit function is obtained by subtracting the total cost (cost per unit times quantity) from total revenue.

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