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 is the integral of the marginal cost function \( C'(Q) \). The marginal cost is given by: \[ C'(Q) = 2e^{0.2Q} \] Integrating with respect to \( Q \), we get: \[ C(Q) = \int 2e^{0.2Q} \, dQ = 10e^{0.2Q} + C_0 \] where \( C_0 \) is the constant of integration.

Step 2: Apply the fixed cost.
We are given that the fixed cost is 75, so when \( Q = 0 \), \( C(0) = 75 \). Thus: \[ C(0) = 10e^{0.2(0)} + C_0 = 75 \] \[ 10 + C_0 = 75 \] \[ C_0 = 65 \]

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