Question

a company finds that the marginal profit, in dollars per foot, from drilling a well that is x feet deep is given by ( p(x)=sqrt4{x} ). find the profit when a well 250 ft deep is drilled
set up the integral for the total profit for a well that is 250 foot deep
( p(250)=int_{0}^{square} square dx )

Explanation:

Step1: Recall the formula for total profit from marginal profit

The total profit \(P(x)\) from a marginal profit function \(P^{\prime}(x)\) is given by the definite - integral \(P(x)=\int_{0}^{x}P^{\prime}(t)dt\). Here, \(x = 250\) and \(P^{\prime}(x)=\sqrt[4]{x}=x^{\frac{1}{4}}\).

Step2: Set up the integral

Substitute \(x = 250\) and \(P^{\prime}(x)\) into the formula. We get \(P(250)=\int_{0}^{250}x^{\frac{1}{4}}dx\).

Answer:

\(P(250)=\int_{0}^{250}x^{\frac{1}{4}}dx\)