Answer:
[tex]P(x) = -10x^2+290x-530[/tex]
Step-by-step explanation:
We are given the following in the question:
[tex]\dfrac{dP}{dx} = -20x + 290[/tex]
Initial condition:
[tex]P(5) = $670[/tex]
Solving the given differential equation, we get,
[tex]\dfrac{dP}{dx} = -20x + 290\\\\dP=(-20x + 290)dx\\\text{Integrating both sides}\\\\\displaystyle\int dP = \int (-20x + 290)dx\\\\P(x) = -20\dfrac{x^2}{2} + 290x + C\\\\\text{where C is constant of integration}\\P(x) = -10x^2+290x + C\\P(5) = 670\\670 = -10(5)^2+290(5) + C\\670 = 1200 + C\\\Rightarrow C = -530\\P(x) = -10x^2+290x-530[/tex]
is the required profit function.
