Question

The number of hours, t, that bacteria spread 10-fold can be modeled by the equation B(t) = B0(10)2t. There are 25 bacteria present initially and a biologist wishes to find out how many hours will elapse until there are 55,000 bacteria present. What is the exact value for the number of hours elapsed, t, in the equation 55,000 = 25(10)2t?

Answer 1

Answer: 1.67 hours.


Explanation:


1) Function that models the number of bacteria:


$B(t)=B_0(10)^{2t}$


2) Bo = 25, then the function is:


$B_0=25(10)^{2t}$


3) To know how many hours will elpase until the number of bactaria is 55,00, you equals B to 55,500 in the equation and solve for t, in this way:


i) start

$55000=25(10)^{2t)$


ii) division property

$2200=10^{2t}$


iii) antilogarithm property

$2t = log_{10}2200$


iv) division property

$t=\frac{log_{10}2200}{2}$


v) Due the operations:


t = 1.67 hours.


Note that the time is less than 2 hours. That sounds fine since after 1 hour there will be 10 times 25 (250 bacteria), and after 2 hours 100 times 25 (2500 bacteria).