Question

If the marginal cost of a firm is given as the function of output, \( C'(Q) = 2e^{0.2Q} \), and if the fixed cost is 75, find the total cost function.

• A. \( 10 e^{0.2Q} + 65 \)
• B. \( 10 e^{0.2Q} \)
• C. \( 10 e^{0.2Q} + 75 \)
• D. \( e^{0.2Q} + 75 \)

Hint:

The total cost is the integral of the marginal cost function, with the constant of integration determined by the fixed cost.

Answer 1

The Correct Option is C

Step 1: Integrate the marginal cost function. The total cost function \( C(Q) \) is obtained by integrating the marginal cost function \( C'(Q) \). Given the marginal cost function as: \[ C'(Q) = 2e^{0.2Q} \] Integrating with respect to \( Q \): \[ C(Q) = \int 2e^{0.2Q} \, dQ = 10e^{0.2Q} + C_0 \] where \( C_0 \) represents the constant of integration.

Step 2: Apply the fixed cost. The fixed cost is provided as 75. This means that when the quantity \( Q \) is 0, the total cost \( C(0) \) is 75. Applying this condition: \[ C(0) = 10e^{0.2(0)} + C_0 = 75 \] \[ 10 + C_0 = 75 \] Solving for \( C_0 \): \[ C_0 = 65 \]

Step 3: Conclusion. The complete total cost function is therefore: \[ C(Q) = 10e^{0.2Q} + 75 \]