Question

Advertising Cost A company that manufactures hydroponic gardening systems estimates that the profit \(P\) (in dollars) from selling a new system is given by \(P=-35 x^{3}+2700 x^{2}-300,000, \quad 0 \leq x \leq 70\) where \(x\) is the advertising expense (in tens of thousands of dollars). Using this model, how much should the company spend on advertising to obtain a profit of $$\$ 1,800,000$$ ?

Short answer

To obtain the profit of $1,800,000, the company should spend on advertising an amount obtained by solving the equation \(-35x^{3} + 2700x^{2} = 2,100,000\), which requires a numerical solution method.

Step-by-step solution

Set the Profit Function Equal to 1,800,000

First, the known profit of $1,800,000 needs to be inserted into the equation \(P = -35x^{3} + 2700x^{2} - 300,000\). This gives the equation: \(-35x^{3} + 2700x^{2} - 300,000 = 1,800,000\).

Simplify the Equation

The equation can be simplified by adding 300,000 to both sides which gives us the new equation: \(-35x^{3} + 2700x^{2} = 2,100,000\).

Solve the Equation

The equation \(-35x^{3} + 2700x^{2} = 2,100,000\) is a cubic equation and it can be solved numerically, for instance, by using the Newton's method or any numerical method using a software or calculator to get the value of \(x\) that satisfies the equation.

Key concepts

Profit Maximization

Profit maximization is a key goal for any business. It involves determining the optimal level of input, such as resources or spending, that yields the highest profit. In the context of this exercise, the company's objective is to find the most effective advertising cost that results in the greatest profit. For our problem, the profit function is \[ P = -35x^{3} + 2700x^{2} - 300,000 \]where \(x\) represents the advertising expense in tens of thousands of dollars. By making smart budgeting and spending decisions, companies can achieve profit maximization.
To find the optimal spending on advertising, it is crucial to analyze the profit equation. In particular, we examine how changes in advertising influence profits, thereby determining the best possible spending level to reach a specified profit target of 1,800,000 after solving the corresponding cubic equation. Breaking the equation down and thoroughly understanding the factors contributing to the profit can guide effective financial planning.

Advertising Costs

Advertising costs are an essential part of any business strategy. They represent the expenditures a company incurs to promote its products or services. In this problem, the company needs to decide the optimal advertising expense to achieve a predetermined profit goal. Here, advertising costs depend on \(x\), expressed in tens of thousands of dollars.

It is important for businesses to evaluate their advertising costs against expected gains. High advertising costs might lead to increased visibility and sales, while too much spending can reduce profit margins. By using models or equations like the cubic equation in the exercise, businesses can precisely calculate the ideal advertising budget that enables them to maximize profits. This calculated approach helps avoid overspending and ensures efficient use of resources.

Numerical Methods

Numerical methods are mathematical tools used to solve equations that might be too complex for algebraic solutions. In this exercise, we deal with a cubic equation, \[ -35x^{3} + 2700x^{2} = 2,100,000 \]which can be challenging to solve analytically. When such an equation arises, numerical methods like Newton's method come into play.

Newton's method is an iterative approach that approximates roots of real-valued functions. It's especially useful for complicated polynomials or when precise solutions aren't feasible with standard techniques. By using software, calculators, or programming techniques, we can obtain accurate and quick solutions for \(x\) in the context of the problem.

• First, provide an initial guess.
• Next, use the derivative of the function to iteratively refine the guess.
• Repeat the process until the solution converges to a satisfactory accuracy.

This method is convenient for business applications, ensuring decisions—like optimal advertising costs—are based on solid mathematical predictions.