Problem 35 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The quarterly profit (in thousands of dollars) of Cunningham Realty is given by $$ P(x)=-\frac{1}{3} x^{2}+7 x+30 \quad(0 \leq x \leq 50) $$ where \(x\) (in thousands of dollars) is the amount of money Cunningham spends on advertising per quarter. a. Find \(P^{\prime}(x)\). b. What is the rate of change of Cunningham's quarterly profit if the amount it spends on advertising is \(\$ 10,000 /\) quarter \((x=10)\) and \(\$ 30,000 /\) quarter \((x=30) ?\)

Step-by-Step Solution

Answer

The derivative of the profit function is \(P'(x) = -\frac{2}{3}x + 7\). For an advertising expense of \$10,000/quarter (x=10), the rate of change of profit is -\$333.33/quarter. For an advertising expense of \$30,000/quarter (x=30), the rate of change of profit is -\$13,000/quarter.

Step 1: Find P'(x)

P(x) is given as \(P(x) = -\frac{1}{3} x^2 + 7x + 30\). To find P'(x), we'll differentiate P(x) with respect to x. Since P(x) is a polynomial, we will use the basic power rule derivative for each term: - The derivative of \(-\frac{1}{3}x^2\) is \(-2*\frac{1}{3}*x^{2-1} = -\frac{2}{3}x\) - The derivative of \(7x\) is \(7\) - The constant term \(30\) has a derivative of \(0\) Now, we'll combine these to find P'(x): $$P'(x) = -\frac{2}{3}x + 7$$

Step2: Find the rate of change in profit for x=10 and x=30

Now we will substitute x=10 and x=30 in the derived expression P'(x) to find the rate of change in the profit for the specified advertising expenses. For x=10: $$P'(10) = -\frac{2}{3}(10) + 7 = -\frac{20}{3} + 7 = -\frac{1}{3}$$ For x=30: $$P'(30) = -\frac{2}{3}(30) + 7 = -\frac{60}{3} + 7 = -20 + 7 = -13$$ Interpretation: - At an advertising expense of \$10,000/quarter (x=10), the rate of change of profit is -\$\dfrac{1}{3}\times1000 = -\$333.33/quarter - At an advertising expense of \$30,000/quarter (x=30), the rate of change of profit is -\$13\times1000 = -\$13,000/quarter These results show that increasing advertising expenses lead to decreasing profits, with a higher decrease when spending more on advertising.

Key Concepts

Polynomial Functions

Polynomial functions play a significant role in many areas, including business and economics. A polynomial function is a mathematical expression involving a sum of powers of variables with coefficients. They can describe various relationships and trends effectively.
In the given exercise, the quarterly profit function, expressed as:

• \(P(x) = -\frac{1}{3} x^2 + 7x + 30 \)

is a quadratic polynomial. Here, \(x\) represents the spending on advertising in thousands of dollars per quarter. The coefficients of this polynomial depict how different levels of advertising spending affect the profits.
When working with polynomial functions in business, they often help model and predict behaviors over certain ranges. They allow businesses to understand how one variable, such as advertising expenditure, impacts another variable, like profit. Understanding the properties of these functions, including their coefficients, is crucial for deriving meaningful insights that can aid in making informed business decisions.

Rate of Change

The rate of change is an essential concept that describes how one quantity changes in relation to another. In the context of the exercise, it refers to how the quarterly profit changes with respect to changes in advertising expenditure.
To find this, we calculate the derivative, \(P'(x)\), of the polynomial function \(P(x)\):

• \(P'(x) = -\frac{2}{3}x + 7\)

By differentiating, we determine the instantaneous rate of change of profit for each dollar spent on advertising. This derivative tells us how much the profit is expected to increase or decrease for each additional thousand dollars spent.
For example, when \(x = 10\) (or when spending \(10,000), the rate of change indicates a slight decrease in profit, while a more significant decrease is noted at \(x = 30\) (or \)30,000). Understanding the rate of change is valuable as it aids businesses in optimizing spending decisions, ensuring resources are allocated effectively to maximize financial outcomes.

Financial Modeling

Financial modeling uses mathematical abstractions to represent how financial dynamics work. In this exercise, the polynomial function representing the quarterly profit model is a form of financial modeling. It helps simulate real-life business situations.
This model provides a way to visualize and interpret how changes in advertising affect profits. With a given model:

• Companies can predict future profits for various advertising budgets.
• Decision-makers gain insights into optimal spending levels or identify areas for cost adjustments.

Through such models, businesses can test different scenarios and strategies before applying them in the real world, providing a better understanding of potential outcomes. A well-structured financial model helps in identifying patterns and strategizing effectively, leading to more informed and strategic business decisions.