Colby and Jack are growing bacteria in an experiment in a laboratory. Colby starts with 50 bacteria in his culture and the number of bacteria doubles every 2 hours. Jack has a different type of bacteria that doubles every 3 hours. How many bacteria should Jack start with so they have the same amount at the end of the day? Show your work.
Let's first calculate the number of bacteria Colby will have at the end of the day.
Since the number of bacteria doubles every 2 hours for Colby, after 24 hours (which is the end of the day), the number of bacteria will double 12 times (24 hours divided by 2 hours).
Starting with 50 bacteria, the number of bacteria at the end of the day will be:
50 * 2^12 = 50 * 4096 = 204,800 bacteria
Now, let's calculate the number of bacteria Jack should start with to have the same amount at the end of the day.
Since the number of bacteria doubles every 3 hours for Jack, after 24 hours (which is the end of the day), the number of bacteria will double 8 times (24 hours divided by 3 hours).
Let's say Jack starts with x bacteria. The number of bacteria at the end of the day will be:
x * 2^8 = x * 256
To have the same amount of bacteria as Colby at the end of the day, the equation would be:
x * 256 = 204,800
Dividing both sides by 256:
x = 204,800 / 256
x = 800
Jack should start with 800 bacteria so they have the same amount at the end of the day.