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: Understanding the Concept:
Marginal cost is the rate of change of the total cost with respect to the number of units produced. In calculus, this is represented by the first derivative of the total cost function, C(x). The marginal cost at a specific production level (x=10) is found by evaluating the derivative at that point.
Step 2: Key Formula or Approach:
Marginal Cost (MC) = \(\frac{dC}{dx}\).
Step 3: Detailed Explanation:
The given total cost function is: \[ C(x) = 0.007x^3 + 26x^2 + 15x + 400 \] First, we find the derivative of C(x) with respect to x to get the marginal cost function, MC(x). \[ MC(x) = \frac{dC}{dx} = \frac{d}{dx}(0.007x^3 + 26x^2 + 15x + 400) \] Using the power rule for differentiation (\(\frac{d}{dx}(ax^n) = anx^{n-1}\)): \[ MC(x) = 0.007(3x^2) + 26(2x) + 15(1) + 0 \] \[ MC(x) = 0.021x^2 + 52x + 15 \] Now, we need to find the marginal cost when 10 items are produced, so we substitute x = 10 into the MC(x) function. \[ MC(10) = 0.021(10)^2 + 52(10) + 15 \] \[ MC(10) = 0.021(100) + 520 + 15 \] \[ MC(10) = 2.1 + 520 + 15 \] \[ MC(10) = 537.1 \] Step 4: Final Answer:
The marginal cost when 10 items are produced is Rs. 537.1.