Question

If a toxin was introduced into a bacteria colony and t hours later the population is given by N(t)=10000(8+t)e^(0.1t). what was the population when the toxin was introduced? when is the population maximized and find the maximum population.

Answer 1

first introduced=80000
there is no max population

Answer 2

its a growth function

Answer 3

i got the first introduced pop but there is a max at t=2 i just dont understand how to get it

Answer 4

what i understand is that for the population to be maximum the tangent of the curve has to equal zero and therefore i need the derivative of this function but i am not sure how to get that

Answer 5

when the tangent of the curve is equal to 0, you will get the critical point of the function which is the global minimum, there is no maximum for this function because the curve always goes up

Answer 6

thats what i thought too but it says the max population is at t=2 and im not sure how they got that

Answer 7

the function is wrong then, it should be\[N(t) = 10000\left( 8+t \right)e^{-0.1t}\]if you take the derivative and set it to 0, you will get t = 2

Answer 8

yes so sorry typo
im having a hard time with the derivative though

Answer 9

\[N(t) = 10000(8 + t)e^{-0.1t}\]use the product rule\[u = 10000(8 + t)\]\[u' = 10000\]\[v = e^{-0.1t}\]\[v' = -0.1e^{-0.1t}\]\[N'(t) = u'v + uv'\]