A laboratory culture contained 5×103 bacteria at 12:00 hours. The population grows at a rate proportional to its size. At 16:00 hours the count is 1.5×104.
a. (i) Form the differential equation governing the population P(t) (where t is the number of hours after 12:00) and obtain the particular function P(t).
b. (ii) How many bacteria will be present at 19:00 hours?
c. (iii) At what time will the population reach 4×104? (Give your answer correct to the nearest minute.)