Elementary Calculus Midterm Report Activity
Elementary Calculus Midtermreportactivity Briefbco116 Elementary Calc
Write a written report describing the various steps of the required calculations regarding the functions, equations, graphing, and differentiation related to a company producing and selling electric pressure cookers. The report should include the formulation of revenue, profit, and cost functions based on given relationships, and determine the production and pricing strategies that maximize revenue and profit under different scenarios, including the imposition of a new tax. Additionally, the report should contain graphical representations of profit maximization and discuss potential operational issues related to scheduling, packaging, and liabilities tied to the company's proposal.
Paper For Above instruction
In analyzing the economic aspects of Company ABC’s pressure cooker production, it is essential to develop mathematical functions that model their operations. This analysis involves understanding the relationships between demand, pricing, costs, and profits, and optimizing these to maximize financial objectives. The methods employed are rooted in elementary calculus, including functions, derivatives, and graphical interpretations.
1. Revenue Function
Given that the price per unit \( P \) is related to quantity \( X \) by \( P = 200 - 0.004 X \), the revenue function \( R(X) \) is obtained by multiplying the unit price \( P \) by the quantity sold \( X \). Therefore:
\[ R(X) = P \times X = (200 - 0.004 X) \times X = 200X - 0.004 X^2 \]
This quadratic function models the total revenue as a function of units sold, capturing how demand responds to pricing.
2. Quantity and Price for Revenue Maximization
To find the production level \( X \) that maximizes revenue, we differentiate \( R(X) \) with respect to \( X \):
\[ R'(X) = 200 - 0.008 X \]
Set the derivative to zero to locate critical points:
\[ 200 - 0.008 X = 0 \]
\[ X = \frac{200}{0.008} = 25,000 \]
Substituting \( X \) back into the demand function to find the corresponding price:
\[ P = 200 - 0.004 \times 25,000 = 200 - 100 = 100\,€ \]
Thus, to maximize revenue, the company should produce 25,000 units and sell them at 100€ each.
3. Profit Function
Fixed costs are €800,000 annually, and variable costs per unit are €40. The total cost \( C(X) \) combines fixed and variable costs:
\[ C(X) = 800,000 + 40X \]
The profit \( \Pi(X) \) is revenue minus costs:
\[ \Pi(X) = R(X) - C(X) = (200X - 0.004 X^2) - (800,000 + 40X) \]
Simplify:
\[ \Pi(X) = 200X - 0.004 X^2 - 800,000 - 40X = (160X) - 0.004 X^2 - 800,000 \]
4. Quantity and Price for Profit maximization
To maximize profit, differentiate \( \Pi(X) \):
\[ \Pi'(X) = 160 - 0.008 X \]
Set to zero:
\[ 160 - 0.008 X = 0 \]
\[ X = \frac{160}{0.008} = 20,000 \]
Find the associated price:
\[ P = 200 - 0.004 \times 20,000 = 200 - 80 = 120\,€ \]
The company should produce 20,000 units, selling at 120€ each, to maximize profit.
5. Graphical Representation
A graph illustrating profit maximization features the profit curve \( \Pi(X) \), displaying a parabola opening downward with its vertex at the maximum point (20,000 units). The axes are labeled appropriately with units, showing the profit in euros against units produced and sold. The point of maximum profit signifies the optimal production volume, guiding managerial decisions.
6. Impact of a New Tax on Cost Function
The €10 tax per product adds to the variable costs, modifying the cost function:
\[ C_{new}(X) = 800,000 + (40 + 10)X = 800,000 + 50X \]
This increases the per-unit costs, affecting profitability calculations.
7. New Profit Function
The revised profit function considering the tax:
\[ \Pi_{new}(X) = R(X) - C_{new}(X) = (200X - 0.004 X^2) - (800,000 + 50X) \]
Simplify:
\[ \Pi_{new}(X) = 150X - 0.004 X^2 - 800,000 \]
This reflects the decreased profitability due to higher costs.
8. Optimal Production and Pricing Under New Conditions
Differentiating the new profit function:
\[ \Pi'_{new}(X) = 150 - 0.008 X \]
Set the derivative to zero:
\[ 150 - 0.008 X = 0 \]
\[ X = \frac{150}{0.008} = 18,750 \]
Corresponding price:
\[ P = 200 - 0.004 \times 18,750 = 200 - 75 = 125\,€ \]
The company should produce approximately 18,750 units and sell at 125€ each to maximize profit after the tax implementation.
Discussion of Operational Considerations
Operationally, scaling production to the optimal levels requires careful scheduling to ensure efficient workforce deployment. Potential issues include workforce availability, machinery capacity, and supply chain logistics to meet the higher production targets. Packaging considerations involve ensuring boxes match the product dimensions, optimize pallet usage, and facilitate safe and cost-effective shipping. Risks associated with the proposal include liabilities such as injury during handling, non-delivery due to supply chain disruptions, and product breakage, which could incur additional costs and legal liabilities. proactive risk management strategies, quality control measures, and logistical planning are essential to mitigate these operational and legal risks.
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