A farmer has got one box of oranges that he is sharing with his neighbors. At the first house, he gives his neighbor half of the box of oranges plus half an orange. At the second house he does the same. He gives his neighbor half of the remaining oranges plus half an orange. He does the same thing at the third, fourth and fifth houses. After the fifth house, he goes home. There are no oranges left in the box. How many oranges were in the box to begin with?

The biologist has a second culture to examine. she knows that the population of the culture doubles every 15 minutes. after 1 hour and 15 minutes, her assistant found that 80,000 bacteria were present. a) what was the size of the initial population? b) predict the size of the culture at t=3 hours. what was the size of the population at 40 minutes? explain and justify your answers. c) create a graph of the population as a function of time. find an equation that can be used to predict the size of the population at any time t. d) examine the rate at which the bacteria culture is growing. how fast is the culture growing after 1 hour? after 1.5 hours? after 2 hours? use a time interval of h = 0.01 hours to estimate these rates. interpret these rates in terms of the context of the problem situation. how do these three rates compare?