Math Problem Statement
Solution
Let's solve the given problem step-by-step.
Problem Details
We are given:
- Cost Function: , which models the total cost of producing units of a product.
- Price Function: , which is the price function for units.
Part (a): Find the Marginal Cost Function
The marginal cost is the derivative of the cost function with respect to . It represents the cost of producing one additional unit.
Calculating the derivative:
So, the marginal cost function is:
Part (b): Set Up an Integral for the Total Revenue Function and Compute the Producer Surplus When 50 Units Are Produced
The total revenue function is given by multiplying the price function by the number of units produced:
To find the producer surplus, we need to integrate the revenue function up to the production level and subtract the total cost.
Producer Surplus is given by:
For , the integral becomes:
- Compute the Integral:
Evaluating from 0 to 50:
= 5000 - 125 = 4875$$ 2. **Calculate the Total Cost $$C(50)$$**: $$C(50) = 50 + 0.01 \times 50^3 = 50 + 0.01 \times 125000 = 50 + 1250 = 1300$$ 3. **Compute Producer Surplus**: $$\text{Producer Surplus} = 4875 - 1300 = 3575$$ ### Final Answer: $$\boxed{\text{Producer Surplus} = 3575}$$ ### Summary - Marginal Cost Function: $$\boxed{C'(x) = 0.03x^2}$$ - Producer Surplus when 50 units are produced: $$\boxed{3575}$$ Would you like any additional details or explanations on these calculations? ### Related Questions: 1. How do you interpret the marginal cost in economic terms? 2. What is the significance of the producer surplus in economic analysis? 3. How would the producer surplus change if the cost function were different? 4. What is the difference between consumer surplus and producer surplus? 5. How would you find the break-even point using cost and revenue functions? ### Tip: To better understand economic functions, always visualize the cost, revenue, and price functions on a graph. This helps in understanding the relationship between these variables visually.
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Math Problem Analysis
Mathematical Concepts
Calculus
Economics
Integration
Marginal Cost
Producer Surplus
Formulas
Marginal Cost = dC(x)/dx
Producer Surplus = ∫(p(x))dx - C(x)
Revenue Function: R(x) = p(x) * x
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Undergraduate - Economics/Calculus
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