Math Problem Statement

A steel plant produces x tons of steel per week at a total cost of Rs. (1/3)x^3 - 5x^2 + 99x + 35. Find the output level at which the marginal cost attains its minimum value. Also find the minimum marginal cost.

Solution

The problem is as follows:

A steel plant produces xx tons of steel per week at a total cost of
C(x)=13x3−5x2+99x+35.C(x) = \frac{1}{3}x^3 - 5x^2 + 99x + 35.
You are asked to:

  1. Find the output level (xx) at which the marginal cost attains its minimum value.
  2. Also find the minimum marginal cost.

Solution:

  1. Marginal Cost:
    The marginal cost is the derivative of the total cost function C(x)C(x) with respect to xx:
    MC(x)=dC(x)dx.MC(x) = \frac{dC(x)}{dx}.

Differentiating C(x)C(x):

= x^2 - 10x + 99.$$ 2. **Find the value of $$x$$ where $$MC(x)$$ is minimized**: To minimize the marginal cost, take the derivative of $$MC(x)$$: $$MC'(x) = \frac{d}{dx}(x^2 - 10x + 99) = 2x - 10.$$ Set $$MC'(x) = 0$$ to find the critical points: $$2x - 10 = 0 \implies x = 5.$$ To confirm it's a minimum, check the second derivative: $$MC''(x) = \frac{d}{dx}(2x - 10) = 2.$$ Since $$MC''(x) > 0$$, the marginal cost is minimized at $$x = 5$$. 3. **Find the minimum marginal cost**: Substitute $$x = 5$$ into $$MC(x)$$: $$MC(5) = (5)^2 - 10(5) + 99 = 25 - 50 + 99 = 74.$$ --- ### Final Answer: 1. The output level at which the marginal cost attains its minimum value is **5 tons**. 2. The minimum marginal cost is **74 Rs**. --- Would you like a detailed breakdown of any step, or do you have further questions? ### Related Questions: 1. What is the importance of minimizing marginal cost in production? 2. How does marginal cost differ from average cost in economics? 3. Could the second derivative test ever fail in such problems? If so, why? 4. How would the total cost function change if additional fixed costs are introduced? 5. How can we determine the range of $$x$$ where marginal cost is increasing? ### Tip: In economics, the marginal cost curve often determines the optimal production level by intersecting with demand or revenue curves.

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Math Problem Analysis

Mathematical Concepts

Calculus
Optimization
Marginal Cost Analysis

Formulas

Marginal cost: MC(x) = dC(x)/dx
Critical points: Set the derivative of MC(x) to zero to find the minimum
Second derivative test to confirm minimum: If MC''(x) > 0, it's a minimum

Theorems

Second Derivative Test

Suitable Grade Level

Grades 11-12 or College Level

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