math

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can you please explain this question to me
for part b i get t=0.7 by trial and error and 86 percent of the population but im sure theres a more exact way to do it

(a) A chemical process produces NaCI at the rate 3t^1/2 grams per minute.
(i) What is the rate of production 4 minutes into the process?
(ii) How much NaCI has been produced by that time?
(b) The rate at which a viral infection spreads through a population is given
by:
r(t) = 2te^-t^2
Where t is measured in months and r(t) is the portion of population
infected per month.
(i) When is the infection spreading most rapidly?
(ii) What portion ofthe population has been infected by this time?

i really need help urgently

  • math -

    r(t)=2t e^-t^2

    r'=0=2e^-t^2 +2te^-t^2 * 2t*-1
    0=2-4t
    t=1/2 month
    check my calculus.
    ii) population=INTEGRAL r(t) dt
    = INT 2te^-t^2 dt

    let u= t^2
    du= 2t dt
    population= int e^-u du=-e^-u
    but u=t^2= 1/4
    population= -e^-u from 0 to 1/4
    population=- e^-1/4 +1= 22percent

    check my work.

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