Question 15 of 82intermediate🔧 ApplyNumerical4 marks

The total cost C(x) in Rupees associated with the production of x units of an item is given by C(x) = 0.007x³ - 0.003x² + 15x + 4000. Find the marginal cost when 17 units are produced.

Correct Answer

Marginal Cost (MC) is the rate of change of total cost with respect to the output. Therefore, we differentiate the cost function C(x)C(x) with respect to xx.

Given: C(x)=0.007x3−0.003x2+15x+4000C(x) = 0.007x^3 - 0.003x^2 + 15x + 4000

Differentiating with respect to xx:

MC=dCdx=0.007(3x2)−0.003(2x)+15MC = \frac{dC}{dx} = 0.007(3x^2) - 0.003(2x) + 15 MC=0.021x2−0.006x+15MC = 0.021x^2 - 0.006x + 15

We need to find the marginal cost when 17 units are produced. Substituting x=17x = 17 in the equation:

MC=0.021(17)2−0.006(17)+15MC = 0.021(17)^2 - 0.006(17) + 15

Calculating the values:

MC=0.021(289)−0.102+15MC = 0.021(289) - 0.102 + 15 MC=6.069−0.102+15MC = 6.069 - 0.102 + 15 MC=5.967+15MC = 5.967 + 15 MC=20.967MC = 20.967

Hence, the marginal cost when 17 units are produced is ₹ 20.97 (nearly).

Exercise: EXERCISE 6.1 | Q: 15 | (Page 5)
For More Understanding

Explanation

This question is solved using the definition of Marginal Cost provided in Example 5 of the textbook context, which states that marginal cost is the rate of change of total cost with respect to the output (MC=dCdxMC = \frac{dC}{dx}). The solution involves differentiating the polynomial cost function and then evaluating the derivative at x=17x = 17. Following the rounding convention shown in Example 5, the final answer is rounded to two decimal places.

Solution Steps

  1. Step 1: Identify the definition of Marginal Cost from the context: MC=dCdxMC = \frac{dC}{dx}.

  2. Step 2: Differentiate the given function C(x)=0.007x3−0.003x2+15x+4000C(x) = 0.007x^3 - 0.003x^2 + 15x + 4000 to get MC=0.021x2−0.006x+15MC = 0.021x^2 - 0.006x + 15.

  3. Step 3: Substitute x=17x = 17 into the derivative: MC=0.021(17)2−0.006(17)+15MC = 0.021(17)^2 - 0.006(17) + 15.

  4. Step 4: Calculate the final value: 6.069−0.102+15=20.9676.069 - 0.102 + 15 = 20.967.

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