population growth

[esmy]

A population of 500 bateria is introduced into a culture and grows in number according to the equation p(t)=500(1+(4t/50+t^2)) where t is measured in hours. find the rate at which the population is growing when t=2. how do i do this????

 

[soroban]

Hello, esmy!

A population of 500 bacteria is introduced into a culture and grows in number
according to the equation: .\(\displaystyle p(t) \:=\:500\left(1+\dfrac{4t}{50+t^2}\right)\) where \(\displaystyle t\) is measured in hours.

Find the rate at which the population is growing when \(\displaystyle t=2.\)

You're in a Calculus class and have never done a Related Rates problem?


\(\displaystyle \text{Differentiate the function with respect to }t\!: \;\;\dfrac{dp}{dt} \;=\;500\cdot\) \(\displaystyle \dfrac{(50+t^2)\!\cdot\!4 - 4t(2t)}{(50+t^2)^2}\)

. . \(\displaystyle \text{which simplifies to: }\:\dfrac{dp}{dt} \;=\;2000\cdot\dfrac{50-t^2}{(50+t^2)^2}\)

\(\displaystyle \text{Now let }t = 2.\)