A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 53 cells.

(a) Find the relative growth rate. (Assume t is measured in hours.)
k=

(b) Find an expression for the number of cells after t hours.
P(t) =

(c) Find the number of cells after 8 hours.

(d) Find the rate of growth after 8 hours. (Round your answer to three decimal places.)

(e) When will the population reach 20,000 cells? (Round your answer to two decimal places.)