ANSWER
t=15 minutes.
EXPLANATION
The equation that models the growth rate of the first type of bacteria is,
[tex]y = 2 {t}^{2} + 3t + 500[/tex]
The growth rate of a second type of bacteria is modeled by:
[tex]y = 3 {t}^{2} + t + 300[/tex]
Since y is the number of bacteria after t minutes in both equations, we equate both equations to find the time when there is an equal number of both types of bacteria.
[tex]3 {t}^{2} + t + 300 = 2 {t}^{2} + 3t + 500[/tex]
We rewrite as standard quadratic equation,
[tex]3 {t}^{2} - 2 {t}^{2} + t - 3t + 300 - 500 = 0[/tex]
[tex] {t}^{2} - 2t - 200 = 0[/tex]
We use the quadratic formula to obtain,
[tex]t = \frac{ - - 2 \pm \sqrt{ {( - 2)}^{2} - 4(1)( - 200) } }{2(1)} [/tex]
[tex]t = \frac{ -2 \pm \sqrt{ 804} }{2} [/tex]
[tex]t = \frac{ -2 \pm 2\sqrt{ 201} }{2} [/tex]
[tex]t = 1 - \sqrt{201} \: or \: t = 1 + \sqrt{201} [/tex]
t=-13.177 or t=15.177
We discard the negative value. Hence there is an equal number of both types of bacteria after approximately 15 minutes.
