Question

Profit A company that produces calculators estimated that the profit \(P\) (in dollars) from selling a particular model of calculator was \(P=-152 x^{3}+7545 x^{2}-169,625, \quad 0 \leq x \leq 45\) where \(x\) was the advertising expense (in tens of thousands of dollars). For this model of calculator, the advertising expense was \(\$ 400,000(x=40)\) and the profit was \(\$ 2,174,375\). (a) Use a graphing utility to graph the profit function. (b) Could the company have obtained the same profit by spending less on advertising? Explain your reasoning.

Short answer

The conclusion depends on the observations made from the graph of the profit function. It will affirm whether the company could have made the same profit (i.e., \$2,174,375) by spending less than \$400,000 on advertising or not.

Step-by-step solution

Graph the profit function

Firstly, input the provided profit function \(P=-152 x^{3}+7545 x^{2}-169,625, \quad 0 \leq x \leq 45\) into a graphing utility. Observe how the graph looks and where the real-world data point \(x=40, P=2,174,375\) falls on the graph.

Analyze the graph for lower advertising cost

Next, look at the graph to understand how the profit changes with different advertising expenses. Look specifically for points on the graph where the profit is equal to \$2,174,375, but the advertising cost \(x\) is less than 40.

Conclude based on observations

If you found points on the graph where the profit is equal to \$2,174,375 with advertising cost \(x\) less than 40, it means the company could have made the same profit by spending less on advertising. If not, then it means the company could not have obtained the same profit with lower advertising expenses. Draw your conclusion based on the observations made from the graph.

Key concepts

Graphing Utility

A graphing utility is a tool that helps visualize mathematical functions. These tools can be physical devices, like graphing calculators, or digital software, such as online graphing platforms and apps.

In this exercise, a graphing utility is used to plot the profit function of a calculator company. The function represents the relationship between advertising expense and profit. To use a graphing utility, you enter the function's equation into the tool. This tool then displays the graph, helping to identify trends and key points on the curve.

Why is a graphing utility useful?

• It simplifies complex equations by visualizing them.
• Helps identify maxima, minima, and points of interest.
• Makes comparison of different points easier, like identifying an advertising expense that yields the needed profit.

In summary, graphing utilities are essential in analyzing polynomial functions when seeking to optimize variables such as advertising costs in relation to profit.

Advertising Expense

Advertising expense refers to the amount of money a company spends to promote its products or services. In this case, it relates to how much the company spends on advertising its calculators. Here, the expense is expressed in tens of thousands of dollars: for instance, an entry `x = 40` equates to $400,000.

The role of advertising expense in profit modeling can be crucial. By analyzing how profit changes alongside the advertising budget, a company can determine the most efficient use of its funds.

Important Considerations:

• Higher spending can potentially increase reach and sales but comes with diminishing returns.
• Identifying the point of maximum profit efficiency helps in budget planning.
• Understanding the elasticity of profit in response to increments in advertising helps in strategic decision-making.

In our scenario, if the profit achieved at x = 40 was demonstrably achievable at a lower cost, this insight could lead to budget optimization.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. The function given in the exercise is a cubic polynomial since the highest exponent of the variable is three.

The function in question, \( P=-152 x^{3}+7545 x^{2}-169,625 \), models the profit based on advertising expenses. It features different terms that reflect how profit changes over varying levels of advertising.

Why are polynomial functions significant in this context?

• They can approximate real-world data due to their flexible nature.
• Polynomials of higher degree like cubic functions can depict more complex relationships.
• Such functions help in evaluating diverse scenarios by assessing different points (or intervals) within the function's domain.

In our case study, using this polynomial, the company seeks to identify an optimal spending strategy to maximize profit without unnecessary expenditure.