Step 1 :The marginal profit is the derivative of the profit function. To find the total profit, we need to integrate the marginal profit function from 0 to 240 feet.
Step 2 :Let's denote the marginal profit function as \(P^{\prime}(x)=x^{\frac{1}{3}}\).
Step 3 :The total profit is then given by the integral of the marginal profit function from 0 to 240, which is \(P(x) = \int_{0}^{240} P^{\prime}(x) dx\).
Step 4 :By calculating the integral, we find that the total profit is approximately 1118.60.
Step 5 :Thus, the company's profit from drilling a well that is 240 feet deep is approximately $1118.60.
Step 6 :Final Answer: \(\boxed{\$ 1,118.60}\)