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

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

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

Answer

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\neq - 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})