Respuesta :
Answer:
It will take 5 hours until it reaches its maximum concentration.
Step-by-step explanation:
The maximum concentration will happen in t hours. t is found when
[tex]C'(t) = 0[/tex]
In this problem
[tex]C(t) = \frac{100t}{t^{2} + 25}[/tex]
Applying the quotient derivative formula
[tex]C'(t) = \frac{(100t)'(t^{2} + 25) - (t^{2} + 25)'(100t)}{(t^{2} + 25)^{2}}[/tex]
[tex]C'(t) = \frac{100t^{2} + 2500 - 200t^{2}}{(t^{2} + 25)^{2}}[/tex]
[tex]C'(t) = \frac{-100t^{2} + 2500}{(t^{2} + 25)^{2}}[/tex]
A fraction is equal to zero when the numerator is 0. So
[tex]-100t^{2} + 2500 = 0[/tex]
[tex]100t^{2} = 2500[/tex]
[tex]t^{2} = 25[/tex]
[tex]t = \pm \sqrt{25}[/tex]
[tex]t = \pm 5[/tex]
We use only positive value.
It will take 5 hours until it reaches its maximum concentration.
