The marginal cost of a product is given by MC = ` (14000)/(sqrt(7x+ 4))` and the fixed cost is 18000.
Find the total cost and the average cost of producing 3 units of output.

Answer

To solve the problem, we need to find the total cost and average cost of producing 3 units of output given the marginal cost function and fixed cost.

### Step 1: Understand the given information
- The marginal cost (MC) is given by:
\[ MC = \frac{14000}{\sqrt{7x + 4}} \]
- The fixed cost (FC) is given as:
\[ FC = 18000 \]

### Step 2: Find the total cost (TC)
The total cost (TC) can be found by integrating the marginal cost with respect to \(x\) and adding the fixed cost.

\[ TC = \int MC \, dx + FC \]

### Step 3: Integrate the marginal cost
We need to integrate:
\[ \int \frac{14000}{\sqrt{7x + 4}} \, dx \]

To perform the integration, we can use the substitution method. Let:
\[ u = 7x + 4 \implies du = 7 \, dx \implies dx = \frac{du}{7} \]

Now, substituting in the integral:
\[ \int \frac{14000}{\sqrt{u}} \cdot \frac{du}{7} = \frac{14000}{7} \int u^{-1/2} \, du = 2000 \cdot 2u^{1/2} = 4000\sqrt{u} \]

Substituting back for \(u\):
\[ = 4000\sqrt{7x + 4} \]

### Step 4: Write the total cost function
Now, we can express the total cost function:
\[ TC = 4000\sqrt{7x + 4} + 18000 \]

### Step 5: Calculate total cost for \(x = 3\)
Now we need to find the total cost when \(x = 3\):
\[ TC = 4000\sqrt{7(3) + 4} + 18000 \]
\[ = 4000\sqrt{21 + 4} + 18000 \]
\[ = 4000\sqrt{25} + 18000 \]
\[ = 4000 \cdot 5 + 18000 \]
\[ = 20000 + 18000 = 38000 \]

### Step 6: Find the average cost (AC)
The average cost (AC) is given by:
\[ AC = \frac{TC}{x} \]
Substituting \(TC\) and \(x = 3\):
\[ AC = \frac{38000}{3} \approx 12666.67 \]

### Final Answers
- Total Cost (TC) for producing 3 units: **38000**
- Average Cost (AC) for producing 3 units: **12666.67**