To find the marginal revenue at a production level of $x = 25$, we need to find the derivative of the revenue function $R(x) = 6x - 0.01x^2$. The derivative represents the rate of change of revenue with respect to the number of units produced, which is the marginal revenue.

Step 1: Differentiate the revenue function

The revenue function is:

$R(x) = 6x - 0.01x^2$

To find the marginal revenue, we differentiate $R(x)$ with respect to $x$:

$\frac{d}{dx} R(x) = \frac{d}{dx} \left( 6x - 0.01x^2 \right)$

The derivative of $6x$ is 6, and the derivative of $-0.01x^2$ is $-0.02x$.

So, the marginal revenue function is:

$R'(x) = 6 - 0.02x$

Step 2: Calculate the marginal revenue at $x = 25$

Now, substitute $x = 25$ into the derivative to find the marginal revenue at this production level:

$R'(25) = 6 - 0.02(25)$ $R'(25) = 6 - 0.5$ $R'(25) = 5.5$

Thus, the marginal revenue at a production level of 25 units is $5.50.

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Would you like more details or have any questions on this? Here are some additional questions that might interest you:

1. What does the marginal revenue represent in terms of business operations?
2. How does the marginal revenue change as production increases or decreases?
3. How can we find the production level where marginal revenue is zero?
4. What is the total revenue when producing 25 units?
5. How would the revenue function change if the production cost per unit increased?

Tip: To find the break-even point (where revenue equals cost), you can set the revenue function equal to the cost function and solve for $x$.