An experimental culture has an initial population of 50 bacteria.The population increased by 80 percent in 20 minutes.Determine the time it will it take to have a population of 1.2 million bacteria?
To determine the time it will take to reach a population of 1.2 million bacteria, we need to find the growth rate per minute and then calculate the time it will take for the population to reach 1.2 million.
The population increased by 80 percent in 20 minutes, so the growth rate per minute can be calculated as:
Growth rate per minute = 80% / 20 minutes = 4% per minute
Let's denote the time it will take for the population to reach 1.2 million bacteria as t minutes.
Using exponential growth formula: Final population = Initial population * (1 + growth rate)^time, we can write:
1.2 million = 50 * (1 + 0.04)^t
Divide both sides of the equation by 50:
24,000 = (1.04)^t
To isolate the exponent, take the logarithm (base 1.04) of both sides:
log base 1.04 (24,000) = log base 1.04 ((1.04)^t)
Simplify the equation:
t = log base 1.04 (24,000)
Using a calculator, we find that log base 1.04 (24,000) ≈ 289.1
Therefore, it will take approximately 289.1 minutes for the population to reach 1.2 million bacteria.
To determine the time it will take for the population to reach 1.2 million bacteria, we need to find the rate of growth and use it to calculate the time needed.
Given:
Initial population = 50 bacteria
Population growth after 20 minutes = 80%
Step 1: Calculate the population growth rate:
Population growth rate = 80% = 0.8
(We convert it to decimal form by dividing the percentage by 100)
Step 2: Calculate the population after 20 minutes:
Population after 20 minutes = Initial population + (Population growth rate * Initial population)
Population after 20 minutes = 50 + (0.8 * 50) = 50 + 40 = 90 bacteria
Step 3: Set up an equation to find the time needed to reach 1.2 million bacteria:
Population after X minutes = 1.2 million bacteria
Step 4: Solve the equation for time (X):
90 * (1 + Population growth rate) ^ (X/20) = 1.2 million
Step 5: Solve for X using logarithms:
(1 + Population growth rate) ^ (X/20) = 1.2 million / 90
(1 + 0.8) ^ (X/20) = 13333.33
Step 6: Take the logarithm of both sides:
(X/20) * log(1.8) = log(13333.33)
Step 7: Solve for X:
X/20 = log(13333.33) / log(1.8)
X = 20 * (log(13333.33) / log(1.8))
Using a calculator, evaluate the right side of the equation to find X:
X ≈ 20 * (9.5006 / 0.2553)
X ≈ 20 * 37.25
X ≈ 745 minutes
Therefore, it will take approximately 745 minutes for the population to reach 1.2 million bacteria.