Questions: The marginal cost of a product is given by 168+225/sqrt(x) dollars per unit, where x is the number of units produced. The current level of production is 51 units weekly. If the level of production is increased to 126 units weekly, find the increase in the total cost. Round your answer to the nearest cent.

The marginal cost of a product is given by 168+225/sqrt(x) dollars per unit, where x is the number of units produced. The current level of production is 51 units weekly. If the level of production is increased to 126 units weekly, find the increase in the total cost. Round your answer to the nearest cent.

Transcript text: The marginal cost of a product is given by $168+\frac{225}{\sqrt{x}}$ dollars per unit, where $x$ is the number of units produced. The current level of production is 51 units weekly. If the level of production is increased to 126 units weekly, find the increase in the total cost. Round your answer to the nearest cent.

Solution

Solution Steps

Step 1: Define the Marginal Cost Function

The marginal cost of the product is given by the function: \[ MC(x) = 168 + \frac{225}{\sqrt{x}} \]

Step 2: Set the Production Levels

We are considering the increase in production from $ x = 51 $ units to $ x = 126 $ units.

Step 3: Integrate the Marginal Cost Function

To find the increase in total cost, we need to integrate the marginal cost function over the interval from 51 to 126: \[ \text{Total Cost Increase} = \int_{51}^{126} \left( 168 + \frac{225}{\sqrt{x}} \right) \, dx \]

Step 4: Evaluate the Integral

The result of the integration is: \[ \text{Total Cost Increase} = -450\sqrt{51} + 1350\sqrt{14} + 12600 \]

Step 5: Calculate the Total Cost Increase

Evaluating the expression gives: \[ \text{Total Cost Increase} \approx 14437.59 \]