Question

The total cost C(x) in Rupees associated with the production of x units of an item is given by C(x) = \(0.007x^3 - 0.003x^2 + 15x + 400\). The marginal cost when 10 items are produced is:

• A. 537.1
• B. 441.15
• C. 1575
• D. 875.25

Hint:

In economics and business calculus, "marginal" almost always means "derivative of". Marginal cost is the derivative of the cost function, marginal revenue is the derivative of the revenue function, and so on. This is a key concept to remember for such application-based problems.

Answer 1

The Correct Option is A

Step 1: Conceptualization:
Marginal cost signifies the incremental expense incurred from producing one additional unit. Mathematically, it is the first derivative of the total cost function, C(x). To determine the marginal cost at a specific output level (e.g., x=10), one evaluates this derivative at that point.
Step 2: Governing Equation:
Marginal Cost (MC) = \(\frac{dC}{dx}\).
Step 3: Derivation and Calculation:
Given the total cost function:\[ C(x) = 0.007x^3 + 26x^2 + 15x + 400 \]The marginal cost function, MC(x), is obtained by differentiating C(x) with respect to x.\[ MC(x) = \frac{dC}{dx} = \frac{d}{dx}(0.007x^3 + 26x^2 + 15x + 400) \]Applying the power rule of differentiation (\(\frac{d}{dx}(ax^n) = anx^{n-1}\)) yields:\[ MC(x) = 0.007(3x^2) + 26(2x) + 15(1) + 0 \]\[ MC(x) = 0.021x^2 + 52x + 15 \]To find the marginal cost for producing 10 units, substitute x = 10 into MC(x).\[ MC(10) = 0.021(10)^2 + 52(10) + 15 \]\[ MC(10) = 0.021(100) + 520 + 15 \]\[ MC(10) = 2.1 + 520 + 15 \]\[ MC(10) = 537.1 \]Step 4: Conclusion:
The marginal cost for producing 10 items is Rs. 537.1.