The generation time G for a particular bacterium is the time it takes for the population to double. The bacteria increase in population is shown by the formula G=t/3.3log_a p, where t is the time period of the population increase, a is the number of bacteria at the beginning of the time period, and P is the number of bacteria at the end of the time period. If the generation time for the bacteria is 4.5 hours, how long will it take 4 of these bacteria to multiply into a colony of 7525 bacteria? Round to the nearest hour.
To find the length of time it will take for 4 bacteria to multiply into a colony of 7525 bacteria, we need to solve for t in the equation G = t/3.3log_a p.
Since G is given as 4.5 hours, we can substitute this value into the equation to get 4.5 = t/3.3log_a p.
The number of bacteria at the beginning of the time period is given as 4, and the number of bacteria at the end of the time period is given as 7525. So we substitute these values into the equation to get 4.5 = t/3.3log_4 7525.
To solve for t, we multiply both sides of the equation by 3.3log_4 7525 to isolate t. This gives us t = 3.3log_4 7525 * 4.5.
Using a calculator to evaluate log_4 7525, we find that log_4 7525 is approximately 5.48. Substituting this value into the equation gives us t = 3.3 * 5.48 * 4.5.
Multiplying these values together gives us t = 80.244. Rounded to the nearest hour, this is approximately 80 hours.
Therefore, it will take approximately 80 hours for 4 bacteria to multiply into a colony of 7525 bacteria.