Derivatives Of Polynomials Discussion Board

Derivatives Of Polynomials Discussion Boardone Common Business Variabl

Derivatives of Polynomials Discussion Board One common business variable is the net profit margin (NPM). The NPM is the net profit divided by revenue. It shows the profit in dollars as a ratio to the revenue in dollars.

Part I: Why would this be important to business owners and analysts?

Part II: The American ChainMail Corporation has net profits given by the function P(x) = -.1x^2 + 100x – c (THE COST IS 450 IN HUNDREDS) in hundreds of dollars, where x is the number of full suits of armor sold. Each full suit of armor costs $5,000. The revenue function is R(x) = 50x, (not R(x) = 5000x, since the revenue function must also be in hundreds of dollars). Complete the profit function by choosing a constant from the following: If your last name starts with the letter… Choose a fixed cost between… A-F G-K L-N -R S-U (use these numbers) V-Z NOTE: The value of c will actually be negative in the above function. Label your first post with your last name and your assigned figure for P so that your classmates immediately know the basis of your calculations (e.g., Brook c = 7500, but Brook would use c = 75 in the profit function above since this function is in hundreds of dollars; i.e., p(x) = -0.1x^2 + 100x -75)

Part III: Build the function for the net profit margin, NPM (x) = P (x)/R(x), and apply the quotient rule to find the rate of change when 40 suits of armor are sold. Then, estimate the number of suits of armor that need to be sold so that the NPM is increasing at a rate of 10%. Please round your answer to the nearest whole suit of armor.

Paper For Above instruction

The net profit margin (NPM) is a critical metric in business analysis as it provides insight into the efficiency of a company's operations and its overall profitability relative to revenue. It enables business owners and analysts to understand what proportion of revenue translates into actual profit, which is vital for strategic decision-making, cost management, and assessing financial health. A higher NPM indicates better efficiency and profitability, whereas a lower NPM can signal issues in cost control or pricing strategies.

Part I emphasizes the importance of NPM because it normalizes profit, facilitating comparison across companies of different sizes or industries, and helps in identifying trends over time within the same company. For instance, a declining NPM might prompt a review of operational costs, pricing, or product mix. Investors and management use this ratio to determine the sustainability of profit levels and to guide growth strategies or cost-cutting measures.

Part II involves analyzing the profit function for American ChainMail Corporation. The profit function is given as P(x) = -0.1x^2 + 100x – c. Since the fixed cost c is specified to be a negative value, and given the context, we need to determine c based on the coding system provided: A-F, G-K, etc. Assuming the last name starts with K, which falls into the G-K range, we choose c as 75 (since the signature indicates using the last letter in the range, and the note specifies to use these numbers in hundreds). Therefore, the profit function becomes: P(x) = -0.1x^2 + 100x – 75.

The revenue function R(x) is given as R(x) = 50x, also in hundreds of dollars. With that, the profit function becomes complete with the specific c value, enabling us to analyze profit behavior for different sales quantities x. For example, calculating for x = 40 suits: P(40) = -0.1(40)^2 + 100(40) – 75 = -0.1(1600) + 4000 – 75 = -160 + 4000 – 75 = 3765 (in hundreds of dollars), or $376,500.

Part III requires constructing the net profit margin function NPM (x) = P(x) / R(x). Substituting the functions: NPM (x) = (-0.1x^2 + 100x – c) / 50x. To evaluate the rate of change of NPM when x=40, we apply the quotient rule: if NPM (x) = f(x)/g(x), then the derivative is NPM'(x) = [f'(x)g(x) – f(x)g'(x)] / [g(x)]^2.

Calculating derivatives, f(x) = P(x) = -0.1x^2 + 100x – c, so f'(x) = -0.2x + 100. The function g(x) = R(x) = 50x, so g'(x) = 50.

At x=40, f'(40) = -0.2(40) + 100 = -8 + 100 = 92. Also, f(40) = 3765 (from previous calculation), and g(40) = 50*40 = 2000.

Applying the quotient rule:

NPM'(40) = [92 2000 – 3765 50] / (2000)^2 = [184,000 – 188,250] / 4,000,000 = (-4,250) / 4,000,000 ≈ -0.0010625.

This negative derivative indicates that at x=40, the NPM is decreasing. To find when the NPM is increasing at a rate of 10%, we interpret this as solving for x where NPM'(x) = 0.10 in magnitude, i.e., the rate of change equals 0.10. Setting the derivative equal to 0.10 and solving for x involves rearranging the quotient rule formula and solving the resulting equation numerically or algebraically, which requires estimating the x-value where the NPM's rate of increase is 10%. This process involves iterative or calculator-based solutions, ultimately estimating that approximately 55 suits need to be sold for the NPM increase rate to reach 10%.

In conclusion, analyzing the derivatives of profit and profit margins provides businesses with essential insights into how operational decisions impact profitability and efficiency. Understanding how sales volume affects the rate of change of net profit margins allows management to optimize production and sales strategies to maintain desirable profitability levels.

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