Math Problem Statement

solve as a business calculus beginner student

If the marginal cost of producing x units of a commodity is given by C'(x) = 0.5x^2 - 3x and the fixed cost is $2,500, find (a) the cost function C(x) and (b) the cost of producing 20 units.

Solution

Let's solve the problem step by step.


Given:

The marginal cost function is: C(x)=0.5x23xC'(x) = 0.5x^2 - 3x Fixed cost is: C(0)=2500C(0) = 2500


Part (a): Find the cost function C(x)C(x):

The cost function C(x)C(x) is obtained by integrating the marginal cost function C(x)C'(x).

C(x)=C(x)dxC(x) = \int C'(x) \, dx

Substitute C(x)=0.5x23xC'(x) = 0.5x^2 - 3x:

C(x)=(0.5x23x)dxC(x) = \int (0.5x^2 - 3x) \, dx

Integrate term by term: C(x)=0.5x333x22+C0C(x) = \frac{0.5x^3}{3} - \frac{3x^2}{2} + C_0

Simplify: C(x)=x363x22+C0C(x) = \frac{x^3}{6} - \frac{3x^2}{2} + C_0

Where C0C_0 is the constant of integration.

From the fixed cost, C(0)=2500C(0) = 2500: C(0)=(0)363(0)22+C0=2500C(0) = \frac{(0)^3}{6} - \frac{3(0)^2}{2} + C_0 = 2500

Thus: C0=2500C_0 = 2500

So the cost function becomes: C(x)=x363x22+2500C(x) = \frac{x^3}{6} - \frac{3x^2}{2} + 2500


Part (b): Find the cost of producing 20 units:

Substitute x=20x = 20 into C(x)C(x): C(20)=(20)363(20)22+2500C(20) = \frac{(20)^3}{6} - \frac{3(20)^2}{2} + 2500

Calculate step by step:

  1. 203=800020^3 = 8000, so 80006=1333.33\frac{8000}{6} = 1333.33
  2. 202=40020^2 = 400, so 3(400)2=600\frac{3(400)}{2} = 600
  3. Add the fixed cost 25002500.

C(20)=1333.33600+2500=3233.33C(20) = 1333.33 - 600 + 2500 = 3233.33


Final Answers:

(a) The cost function is: C(x)=x363x22+2500C(x) = \frac{x^3}{6} - \frac{3x^2}{2} + 2500

(b) The cost of producing 20 units is: C(20)=3233.33C(20) = 3233.33


Do you have questions or need more clarification? Let me know!


5 Related Questions:

  1. What is the interpretation of marginal cost in business terms?
  2. How do fixed costs impact the total cost function?
  3. Can marginal costs ever decrease, and how would that affect C(x)C(x)?
  4. How does integrating marginal cost differ from differentiating total cost?
  5. What happens if fixed costs are reduced to zero in this problem?

Tip:

When solving for a cost function from marginal cost, always remember to add the fixed cost at the end. This ensures the total cost function is accurate.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Integration
Marginal Cost Analysis

Formulas

C(x) = ∫C'(x) dx + C₀
C'(x) = 0.5x^2 - 3x
C(x) = x^3 / 6 - 3x^2 / 2 + C₀

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

College Level or Advanced High School (Business Calculus)