Colby and jaquan are growing bacteria in an experiment in a laboratory. colby starts with 50 bacteria in his culture and the number of bacteria doubles every two hours. jaquan has a different type of bacteria that doubles every three hours. how many bacteria should jaquan start with so that they have they have the same amount at the end of the day?

To get started, we will use the general formula for bacteria growth/decay problems:



where: 
A_{f} = Final amount
A_{i} = Initial amount
k = growth rate constant
t = time

For doubling problems, the general formula can be shortened to:



Now, we can use the shortened formula to calculate the growth rate constant of both bacteria:

Colby (1):

per hour

Jaquan (2):

per hour

Using Colby's rate constant, we can use the general formula to calculate for Colby's final amount after 1 day (24 hours).

Note: All units must be constant, so convert day to hours.




Remember that the final amount for both bacteria must be the same after 24 hours. Again, using the general formula, we can calculate the initial amount of bacteria that Jaquan needs:

Answer

1 number

explanation

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

 

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.