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: Concept Definition:
Marginal cost signifies the increase in total cost resulting from the production of one additional unit. Mathematically, it is the first derivative of the total cost function, C(x). To determine the marginal cost at a specific production quantity (e.g., x=10), we evaluate this derivative at that point.
Step 2: Governing Formula:
Marginal Cost (MC) = \(\frac{dC}{dx}\).
Step 3: Calculation Process:
Given the total cost function:\[ C(x) = 0.007x^3 + 26x^2 + 15x + 400 \]First, differentiate C(x) with respect to x to derive the marginal cost function, MC(x).\[ MC(x) = \frac{dC}{dx} = \frac{d}{dx}(0.007x^3 + 26x^2 + 15x + 400) \]Applying the power rule of differentiation (\(\frac{d}{dx}(ax^n) = anx^{n-1}\)):\[ MC(x) = 0.007(3x^2) + 26(2x) + 15(1) + 0 \]\[ MC(x) = 0.021x^2 + 52x + 15 \]Next, substitute x = 10 into the MC(x) function to find the marginal cost for producing 10 items.\[ MC(10) = 0.021(10)^2 + 52(10) + 15 \]\[ MC(10) = 0.021(100) + 520 + 15 \]\[ MC(10) = 2.1 + 520 + 15 \]\[ MC(10) = 537.1 \]Step 4: Conclusive Result:
The marginal cost for producing 10 items amounts to Rs. 537.1.