can you please explain this question to me

for part b i get t=0.7 by trial and error and 0.6 percent of the population but im sure theres a more exact way to do it

(a) A chemical process produces NaCI at the rate 3.Jtgrams 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:
2 r(t) = 2te- t
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?

thank so much

Sure! Let's go through each part of the question step by step and explain how to get the answers.

(a)
(i) The rate of production 4 minutes into the process can be found by substituting t = 4 into the given equation:
rate(t) = 3t grams per minute.
So, rate(4) = 3(4) = 12 grams per minute.

(ii) To calculate the total amount of NaCI produced by that time, we need to integrate the rate of production function with respect to time.
The integral of rate(t) = 3t gives us the total amount of NaCI produced.
Integrating, we get:
∫(3t) dt = (3/2)t^2 + C,
where C is the constant of integration.
To find the amount produced at t = 4, we substitute the values in the equation:
Amount produced = (3/2)(4^2) + C.
Since the initial amount produced is not given in the question, we can't determine the constant C or the absolute amount produced.

(b)
(i) To find when the infection is spreading most rapidly, we need to find the maximum value of the rate function r(t). In this case, the rate function is given by:
r(t) = 2te^(-t).
To find when the rate is maximum, we need to find the critical points of the function.
We can do this by taking the derivative of r(t) and setting it equal to zero:
r'(t) = 2e^(-t) -2te^(-t) = 0.
Simplifying, we get:
2e^(-t)(1 - t) = 0.
This equation holds when either e^(-t) = 0 (which is not possible) or (1 - t) = 0.
Solving for t, we find that t = 1.
Therefore, the infection is spreading most rapidly at t = 1 month.

(ii) To determine the portion of the population infected by a certain time, we need to integrate the rate function r(t) with respect to time from 0 to that certain time.
Integrating, we get:
∫(2te^(-t)) dt = -2te^(-t) -2e^(-t) + C,
where C is the constant of integration.
To find the portion of the population infected at a specific time, substitute the time value into the equation and simplify. However, without knowing the specific time given in the question, we can't determine the exact portion of the population infected.

I hope this explanation helps you understand how to approach and solve these types of problems. Let me know if you have any further questions!