英语翻译abacteriacultureinitiallycontains100cellsandgrowsatarateproportionaltoitssize.afteranhourthepopulationhasincreasedto420.Findanexpressionforthenumberofbacteriaaftert-hours?

Question

英语翻译
a bacteria culture initially contains 100 cells and grows at a rate proportional to its size.after an hour the population has increased to 420.
Find an expression for the number of bacteria after t-hours?

Answer 1

1个发酵培养基最初包含100个细胞,细胞的增长速度和它的个数成正比.
已知,一个小时后细胞的个数已增至420个.
找到一个t小时后,细菌个数的表达式.
设f(t)是所求的表达式,t > 0.
[ Let f(t) be the expression,t > 0]
f(0) = 100,f(1) = 420
f'(t) = af(t),
其中,a为常数.a > 0.
[where,a is a const,a > 0]
d[f(t)]/f(t) = adt,
ln|f(t)| = at + c,
|f(t)| = e^[at + c]
f(t) = de^[at],in which,d is a const.
100 = f(0) = de^[0] = d,
420 = f(1) = de^[a],e^a = 420/d = 420/100 = 4.2,a = ln(4.2)
f(t) = 100e^[ln(4.2)t] = 100[e^ln(4.2)]^t = 100(4.2)^t,t > 0.