A manufacturer's marginal cost function is `(500)/sqrt(2x+25)`. Find the cost involved to increase production from 100 units to 300units.

Answer

To find the cost involved in increasing production from 100 units to 300 units given the marginal cost function \( MC(x) = \frac{500}{\sqrt{2x + 25}} \), we will follow these steps:

### Step 1: Understand the Marginal Cost Function
The marginal cost function represents the cost of producing one additional unit. To find the total cost for increasing production from 100 to 300 units, we need to integrate the marginal cost function.

### Step 2: Set Up the Integral
The total cost \( C \) from producing \( x_1 \) to \( x_2 \) can be found using the integral of the marginal cost function:
\[ C = \int_{x_1}^{x_2} MC(x) \, dx = \int_{100}^{300} \frac{500}{\sqrt{2x + 25}} \, dx \]

### Step 3: Perform a Substitution
Let \( t = 2x + 25 \). Then, differentiate to find \( dx \):
\[ dt = 2 \, dx \quad \Rightarrow \quad dx = \frac{dt}{2} \]
Now, we also need to change the limits of integration:
- When \( x = 100 \), \( t = 2(100) + 25 = 225 \)
- When \( x = 300 \), \( t = 2(300) + 25 = 625 \)

### Step 4: Rewrite the Integral
Substituting \( t \) and \( dx \) into the integral gives:
\[ C = \int_{225}^{625} \frac{500}{\sqrt{t}} \cdot \frac{dt}{2} = 250 \int_{225}^{625} t^{-\frac{1}{2}} \, dt \]

### Step 5: Integrate
The integral of \( t^{-\frac{1}{2}} \) is:
\[ \int t^{-\frac{1}{2}} \, dt = 2t^{\frac{1}{2}} + C \]
Thus,
\[ C = 250 \left[ 2t^{\frac{1}{2}} \right]_{225}^{625} = 250 \left[ 2\sqrt{625} - 2\sqrt{225} \right] \]

### Step 6: Calculate the Values
Now calculate \( \sqrt{625} \) and \( \sqrt{225} \):
\[ \sqrt{625} = 25, \quad \sqrt{225} = 15 \]
Substituting these values back into the equation:
\[ C = 250 \left[ 2(25) - 2(15) \right] = 250 \left[ 50 - 30 \right] = 250 \times 20 = 5000 \]

### Final Answer
The cost involved to increase production from 100 units to 300 units is **5000**.

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