At 3a.m., there is a colony of 736 bacteria, and it is growing at a continuous hourly rate of 10.77%. How many bacteria are there at 7p.m.? Round your answer to the nearest whole number of bacteria.
To find the number of bacteria at 7p.m., we need to calculate the number of hours between 3a.m. and 7p.m. We then use the continuous hourly growth rate to calculate the total growth over that time period.
Step 1: Calculate the number of hours between 3a.m. and 7p.m.
There are 12 hours between 3a.m. and 3p.m., and another 7 hours between 3p.m. and 7p.m. So, the total number of hours is 12 + 7 = 19 hours.
Step 2: Calculate the growth rate per hour.
The growth rate is 10.77%, which means that for every hour that passes, the number of bacteria will increase by 10.77% of its current value. To calculate the growth rate per hour in decimal form, we divide the growth rate percentage by 100: 10.77/100 = 0.1077.
Step 3: Calculate the number of bacteria at 7p.m.
To find the number of bacteria at 7p.m., we need to multiply the initial number of bacteria (736) by the growth rate per hour raised to the power of the total number of hours: 736 * (1 + 0.1077)^19 = 736 * 1.1077^19.
Using a calculator, we find that 1.1077^19 ≈ 3.64 (rounded to two decimal places).
So, multiplying 736 by 3.64, we get 2673.44.
Step 4: Round the answer to the nearest whole number.
Rounding to the nearest whole number, we get 2673 bacteria at 7p.m.