Question

find the function p that satisfies the following conditions (assume x > 0): p(x)=\frac{50}{x^{3}}, p(6)=4. p(x)=

Explanation:

Step1: Integrate \(p'(x)\)

We know that if \(p'(x)=\frac{50}{x^{3}}\), we can rewrite it as \(p'(x) = 50x^{- 3}\). Integrating using the power - rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), we have \(p(x)=\int50x^{-3}dx=50\times\frac{x^{-3 + 1}}{-3 + 1}+C=-25x^{-2}+C=-\frac{25}{x^{2}}+C\).

Step2: Use the condition \(p(6)=4\)

Substitute \(x = 6\) and \(p(6)=4\) into \(p(x)=-\frac{25}{x^{2}}+C\). So \(4=-\frac{25}{6^{2}}+C\). Then \(4=-\frac{25}{36}+C\). Solving for \(C\), we get \(C = 4+\frac{25}{36}=\frac{144 + 25}{36}=\frac{169}{36}\).

Answer:

\(p(x)=-\frac{25}{x^{2}}+\frac{169}{36}\)