1. The problem statement, all variables and given/known data Microbiologists contribute their expertise to many fields, including medicine,environmental science, and biotechnology. Enumerating, the process of countingbacteria, allows microbiologists to build mathematical models that predict populations after a given amount of time has elapsed. Once they can predicta population accurately, the model can be used in medicine, for example, topredict the dose of medication required to kill a certain bacterial infection. The data in the table shown was used by a microbiologist to produce a polynomial-based mathematical model to predict population p (t) as a function of time t, in hours, for the growth of acertain strain of bacteria: p (t) = 1000( 1 + t + 1/2 t^2 + 1/6 t^3 + 1/24 t^4 + 1/120 t^5 ) time (h)- population 0.0-- 1000 0.5-- 1649 1.0-- 2718 1.5-- 4482 2.0-- 7389 How well does the function fit the data? Use the data, the equation, a graph, and/or a graphing calculator to comment on the “goodness of fit.” Use p(t) and p'(t) to determine the following: a)the population after 0.5 h and the rate at which the population is growing at this time. b)the population after 1.0 h and the rate at which the populationis growing at this time. What pattern did you notice in your calculations? Explain this pattern by examining the terms of the equation to find the reason why 2. Relevant equations p (t) = 1000( 1 + t + 1/2 t^2 + 1/6 t^3 + 1/24 t^4 + 1/120 t^5 ) 3. The attempt at a solution i subbed each time value in the p(t) equation and i got the population (it matches the given table) i don't know how to explain this. i got the derivative of p(t) but i'm not sure if it is right. can someone post the derivative of p (t).