[填空题]
There is a rumour spreadi f.6ehar m1 ra3bs,g7nng about thj68eq :;kc; nu ygalhjk(f .m5e questions that will appear in an upcoming chemistry exam in a class with a large number of students. Let x be the proportion of studenk;;u a 8qnjlj(6fgm.: hye k5cts who have heard the rumor and let t be the time in hours, after 10.00 a.m.
The situation can be modelled by the differential equation $ \frac{\mathrm{d} x}{\mathrm{~d} t}=k x(1-x)$ where k is a constant.
1. Use partial fractions to solve this differential equation and hence show that $\frac{x}{1-x}=A e^{k t} $, where A is a constant
2. At 10.00 $\mathrm{a}$ .$ \mathrm{m}$ . one tenth of the students know about the rumour. Find the value of A
3. At 12.00 p.m., the proportion of students who knew about the rumor is 0.55 . Find the value of k ≈
4. Hence, find the proportion of students who knew about the rumour at 1.00 p.m.≈
