Step 1 :The company's marginal cost function is given by \(\frac{15}{\sqrt{x}}\) where \(x\) is the number of units.
Step 2 :The total cost is the integral of the marginal cost function from 0 to 64.
Step 3 :We need to calculate this integral.
Step 4 :By calculating the integral, we find that the total cost is 240.
Step 5 :Final Answer: The total cost of the first 64 units is \(\boxed{240}\).