IF the marginal cost is defined as the rate of change of total cost with respect to the number of units of the product, The marginal cost of producing x units of a product is given by marginal cost `=2xsqrt(x+5)`. The cost of producing 4 units of the product is Rs 314.40 . Find the cost function.

Answer

To find the cost function given the marginal cost, we will follow these steps:

### Step 1: Understand the relationship between marginal cost and total cost
The marginal cost (MC) is defined as the derivative of the total cost (C) with respect to the number of units produced (x):
\[ MC = \frac{dC}{dx} \]
Given that the marginal cost is \( MC = 2x \sqrt{x + 5} \).

### Step 2: Set up the equation for total cost
We can express the relationship as:
\[ \frac{dC}{dx} = 2x \sqrt{x + 5} \]

### Step 3: Integrate to find the total cost function
To find the total cost function \( C \), we need to integrate the marginal cost:
\[ C = \int 2x \sqrt{x + 5} \, dx \]

### Step 4: Use substitution for integration
Let \( t = x + 5 \). Then, \( dt = dx \) and \( x = t - 5 \). When \( x = 0 \), \( t = 5 \).

Now, substituting into the integral:
\[ C = \int 2(t - 5) \sqrt{t} \, dt \]
This can be simplified to:
\[ C = \int 2(t^{3/2} - 5\sqrt{t}) \, dt \]

### Step 5: Integrate term by term
Now, we integrate each term separately:
\[ C = 2 \left( \frac{2}{5} t^{5/2} - 5 \cdot \frac{2}{3} t^{3/2} \right) + k \]
\[ C = \frac{4}{5} t^{5/2} - \frac{20}{3} t^{3/2} + k \]

### Step 6: Substitute back for \( t \)
Now substitute back \( t = x + 5 \):
\[ C = \frac{4}{5} (x + 5)^{5/2} - \frac{20}{3} (x + 5)^{3/2} + k \]

### Step 7: Use the given condition to find \( k \)
We know that the cost of producing 4 units is Rs 314.40. So, we substitute \( x = 4 \):
\[ 314.40 = \frac{4}{5} (4 + 5)^{5/2} - \frac{20}{3} (4 + 5)^{3/2} + k \]
Calculating \( (4 + 5)^{5/2} = 9^{5/2} = 243 \) and \( (4 + 5)^{3/2} = 9^{3/2} = 27 \):
\[ 314.40 = \frac{4}{5} \cdot 243 - \frac{20}{3} \cdot 27 + k \]
Calculating the terms:
\[ \frac{4}{5} \cdot 243 = 194.4 \]
\[ \frac{20}{3} \cdot 27 = 180 \]
So,
\[ 314.40 = 194.4 - 180 + k \]
\[ 314.40 = 14.4 + k \]
Thus,
\[ k = 314.40 - 14.4 = 300 \]

### Step 8: Write the final cost function
Now substituting \( k \) back into the cost function:
\[ C = \frac{4}{5} (x + 5)^{5/2} - \frac{20}{3} (x + 5)^{3/2} + 300 \]

### Final Answer
The cost function is:
\[ C(x) = \frac{4}{5} (x + 5)^{5/2} - \frac{20}{3} (x + 5)^{3/2} + 300 \]