## To find the cost function C(x), we need to integrate the marginal cost function C'(x) with respect to x.

The given marginal cost function is:

C'(x) = 50/sqrt(x)

To integrate this function, we can apply the power rule of integration. The power rule states that integrating x^k with respect to x results in (x^(k+1))/(k+1), where k is any real number except -1.

In this case, we have an exponent of -1/2 in the denominator of the marginal cost function. Adding 1 to the exponent gives us -1/2 + 1 = 1/2.

Let's integrate the function:

âˆ«(50/sqrt(x)) dx

= âˆ«(50 * x^(-1/2)) dx

Applying the power rule, we get:

= 50 * (x^(1/2))/(1/2) + C

Simplifying further, we have:

= 100 * sqrt(x) + C

Now, we need to include the fixed cost, which is $25,000.

This fixed cost represents the cost at x=0.

So, we can write our cost function C(x) as:

C(x) = 100 * sqrt(x) + 25,000

Therefore, the cost function C(x) is given by 100 * sqrt(x) + 25,000.