Question

The population P(t) of a culture of the pseudomonas aeruginosa is given by P(t) = -1709t^2 + 80,000t + 10,000, where t is the time in hours since the culture was started. What is the maximum?

Answer 1

Check the picture below.

so the path of the population P(t) is parabolic, more or less like the one in the picture, so it reaches its maximum at the vertex and at "t" time of the x-coordinate of the vertex.

`\textit{vertex of a vertical parabola, using coefficients} \\\\ P(t)=\stackrel{\stackrel{a}{\downarrow }}{-1709}t^2\stackrel{\stackrel{b}{\downarrow }}{+80000}t\stackrel{\stackrel{c}{\downarrow }}{+10000} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right)`

`\left(-\cfrac{ 80000}{2(-1709)}~~~~ ,~~~~ 10000-\cfrac{ (80000)^2}{4(-1709)}\right) \implies \left( - \cfrac{ 80000 }{ -3418 }~~,~~10000 - \cfrac{ 6400000000 }{ -6836 } \right) \\\\\\ \left( \cfrac{ -40000 }{ -1709 } ~~~~ ,~~~~ 10000 + \cfrac{ 1600000000 }{ 1709 } \right) ~~ \approx ~~ (\stackrel{ hours }{\text{\LARGE 23}}~~,~~946220)`