Math Problem Statement

An economist uses the cost function C(x) = 50 + 0.01x^3 to model the total cost of producing x units of a product. a) Find the marginal cost function. b) Suppose the price function is p(x) = 100 - 0.1x. Set up an integral for the total revenue function and compute the producer surplus when 50 units are produced.

Solution

Let's solve the given problem step-by-step.

Problem Details

We are given:

  • Cost Function: C(x)=50+0.01x3C(x) = 50 + 0.01x^3, which models the total cost of producing xx units of a product.
  • Price Function: p(x)=1000.1xp(x) = 100 - 0.1x, which is the price function for xx units.

Part (a): Find the Marginal Cost Function

The marginal cost is the derivative of the cost function C(x)C(x) with respect to xx. It represents the cost of producing one additional unit.

Marginal Cost C(x)=ddx(50+0.01x3)\text{Marginal Cost } C'(x) = \frac{d}{dx}(50 + 0.01x^3)

Calculating the derivative:

C(x)=0+0.013x2=0.03x2C'(x) = 0 + 0.01 \cdot 3x^2 = 0.03x^2

So, the marginal cost function is:

C(x)=0.03x2\boxed{C'(x) = 0.03x^2}

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:

R(x)=p(x)x=(1000.1x)x=100x0.1x2R(x) = p(x) \cdot x = (100 - 0.1x)x = 100x - 0.1x^2

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:

Producer Surplus=0xp(q)dqC(x)\text{Producer Surplus} = \int_{0}^{x} p(q) \, dq - C(x)

For x=50x = 50, the integral becomes:

Producer Surplus=050(1000.1q)dqC(50)\text{Producer Surplus} = \int_{0}^{50} (100 - 0.1q) \, dq - C(50)

  1. Compute the Integral:

(1000.1q)dq=100q0.05q2\int (100 - 0.1q) \, dq = 100q - 0.05q^2

Evaluating from 0 to 50:

[100q0.05q2]050=(100×500.05×502)(100×00.05×02)\left[100q - 0.05q^2 \right]_{0}^{50} = (100 \times 50 - 0.05 \times 50^2) - (100 \times 0 - 0.05 \times 0^2)

= 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