Maximizing Profit The weekly demand for the Pulsar 40-in....

Maximizing Profit The weekly demand for the Pulsar 40-in. high-definition television is given by the demand equation $$p=-0.05 x+600 \qquad(0 \leq x \leq 12,000)$$ where $p$ denotes the wholesale unit price in dollars and $x$ denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by $$C(x)=0.000002 x^{3}-0.03 x^{2}+400 x+80,000$$ where $C(x)$ denotes the total cost incurred in producing $x$ sets. Find the level of production that will yield a maximum profit for the manufacturer. Hint: Use the quadratic formula.

Maximizing Profit The weekly demand for the Pulsar 40-in. high-definition television is given by the demand equation
$$p=-0.05 x+600 \qquad(0 \leq x \leq 12,000)$$
where $p$ denotes the wholesale unit price in dollars and $x$ denotes the quantity demanded. The weekly total cost function associated with manufacturing these sets is given by
$$C(x)=0.000002 x^{3}-0.03 x^{2}+400 x+80,000$$
where $C(x)$ denotes the total cost incurred in producing $x$ sets. Find the level of production that will yield a maximum profit for the manufacturer.
Hint: Use the quadratic formula.

Key Concepts

Quadratic Formula

The quadratic formula is a method used to solve quadratic equations, which are equations of the second degree. In many optimization problems, including profit maximization, the process of setting the derivative of the profit function equal to zero results in a quadratic equation. The quadratic formula then helps determine the critical points where maximum or minimum values might occur.

Optimization Techniques

Optimization techniques involve using mathematical methods to determine the best outcome (such as maximum profit or minimum cost) under a given set of constraints. Techniques such as differentiation to find critical points and analyzing the second derivative to ensure a maximum or minimum are widely applied in problems across economics and operational research.

Profit Function

The profit function is derived by subtracting the cost function from the revenue function. It models the net income from production and sales, and is used to identify the production level that yields the highest profit. This function is fundamental in optimizing business performance.

Cost Function

A cost function details the total expense incurred in producing a given number of items. It encompasses fixed and variable costs, and is critical for understanding how production costs vary with output levels. Analyzing the cost function allows businesses to make informed decisions about production and pricing strategies.

Revenue Function

A revenue function represents the total income a company earns from selling a certain quantity of goods or services, usually expressed as the product of the price per unit and the number of units sold. It is essential in determining overall profitability, especially when price varies with demand.

Profit Maximization

Profit maximization is the process of determining the level of output or production at which a firm's profit is at its highest. This involves analyzing the relationship between revenue and cost, and finding the point where the difference between the two is greatest. It is a core concept in business and economics, ensuring that resources are allocated efficiently to maximize financial returns.

Please give Ace some feedback