[tex]P'(x)=50-0.10x[/tex]
[tex]\implies P(x)=\displaystyle\int(50-0.10x)\,\mathrm dx[/tex]
[tex]P(x)=50x-\dfrac{0.10}2x^2+C[/tex]
[tex]P(x)=50x-0.05x^2+C[/tex]
Given that [tex]P(0)=-500[/tex], we have
[tex]-500=50(0)-0.05(0)^2+C\implies C=-500[/tex]
and so
[tex]P(x)=50x-0.05x^2-500[/tex]
which means that
[tex]P(200)=50(200)-0.05(200)^2-500[/tex]
[tex]P(200)=7500[/tex]
