Raul enter the hospital at 9:00 am with an infection and has 327,680 bacteria in his body.

Question

1a.) the bacteria in his blood are doubling every 42 minutes in the lab.

 *Write and exponential equation to represent the total bacteria in Raul's body. let x represent time in 42 minute intervals.
*Find the number of times the bacteria have doubled when the bacteria have increased to 10,485,760

1b.)  If Raul's condition remains unchanged then find:
*How long it will be until Raul has 1 billion bacteria in his body?

* determine how long ago Raul was infected before he initially contracted 10 bacteria

2a.) Keleb and Raul have the same infection. the lab has found a medicine that kills 1/3 of bacteria with every dose. the number of bacteria in Kelebs body in represented by the expression : 12,680 (2/3)x where x represent the number of treatments.

*how many bacteria will there be after 7 treatments?
*show the computations you used to solve the problem.

2b.) the doctor plans to treat kaleb at the hospital until his bacteria count is less than 0.5

*How many treatments are needed to reach this goal?
*include at least two computations that show how you arrived at your solution.

Answer 1

1a)

B(x) = 327680(2)^x

10485760 = 327680(2)^x

2^x = 10485760 / 327680 = 32

x = 5 therefore the bacteria have doubled 5 times.

 

1b)

1000000000 = 327680(2)^x

2^x = 1000000000 / 327680 ~= 3051.7578125

x ~= ln(3051.758) / ln(2) ~= 11.575

Since each x represents a 42 minute interval we can find the total time by multiplying 11.575 * 42 ~= 486 minutes.

 

2a)

Simply plug in 7 for x.

 

2b)

0.5 = 12680(2/3)^x

(2/3)^x = 0.5 / 12680

x = ln(0.5/12680) / ln(2/3) ~= 25.01 therefore 26 treatments will be necessary (because 25 is just short of the goal).