4. The total cost of producing \( x \) kllos of a product ls given by \( C=40+100 \sqrt{x} \) (a) What will be the margioal cost, when the output is 1,600 kilos? (b) What will be the output, when marginal cost is Rs, 0,50 per kllo?

To solve the problem, we need to find the marginal cost from the total cost function given by \[ C = 40 + 100 \sqrt{x} \] ### Step 1: Find the Marginal Cost Function The marginal cost (MC) is the derivative of the total cost (C) with respect to the quantity produced (x). Let's calculate the derivative \( \frac{dC}{dx} \): \[ C = 40 + 100 \sqrt{x} \] Taking the derivative: \[ \frac{dC}{dx} = 0 + 100 \cdot \frac{1}{2\sqrt{x}} = \frac{50}{\sqrt{x}} \] So, the marginal cost function is: \[ MC = \frac{50}{\sqrt{x}} \] ### Step 2: Calculate the Marginal Cost when \( x = 1600 \) Now, we will substitute \( x = 1600 \) into the marginal cost function: \[ MC = \frac{50}{\sqrt{1600}} \] Calculating \( \sqrt{1600} \): \[ \sqrt{1600} = 40 \] Now substituting back: \[ MC = \frac{50}{40} = 1.25 \] ### Step 3: Find the Output when Marginal Cost is Rs. 0.50 per Kilo We need to find \( x \) when \( MC = 0.50 \): \[ 0.50 = \frac{50}{\sqrt{x}} \] To solve for \( x \), we can rearrange the equation: \[ \sqrt{x} = \frac{50}{0.50} = 100 \] Now squaring both sides: \[ x = 100^2 = 10000 \] ### Summary of Results (a) The marginal cost when the output is 1,600 kilos is Rs. 1.25. (b) The output when the marginal cost is Rs. 0.50 per kilo is 10,000 kilos.