The bacteria in a 8-liter container double every 2 minutes. After 55 minutes the container is full. How long did it take to fill a quarter of the container?
55-2+54 min --1/2 bacteria
54-2=51 min --1/4 bzcteria
Let's break down the problem step by step:
1. We know that the bacteria in the container double every 2 minutes.
2. After 55 minutes, the container is full.
3. We need to find out how long it took to fill a quarter of the container.
To solve this problem, we can set up a proportion:
Let x represent the number of minutes it takes to fill a quarter of the container.
According to the information given, we can set up the following equation:
(1/4) * 8 liters = 1 liter
Let's find out how much the bacteria multiply in x minutes:
(1 liter) * (2^(x/2)) = 8 liters
Simplifying this equation, we get:
2^(x/2) = 8
To solve for x, we take the logarithm of both sides:
log base 2 of (2^(x/2)) = log base 2 of 8
(x/2) * log base 2 of 2 = log base 2 of 8
Since log base 2 of 2 is equal to 1, we have:
x/2 = log base 2 of 8
x/2 = 3
Multiplying both sides by 2, we get:
x = 6
Therefore, it took 6 minutes to fill a quarter of the container.
To find out how long it took to fill a quarter of the container, we need to calculate the time it took for the bacteria to multiply and fill the entire container first.
Given that the bacteria double every 2 minutes, we can use exponential growth to determine the number of bacteria at each time interval.
Let's denote the starting number of bacteria as N (which is 1, because bacteria grow from one individual). After 2 minutes, the number of bacteria will be 2N (since they double every 2 minutes). After 4 minutes, it will be 2 * 2N = 2^2 * N. Similarly, after 6 minutes, it will be 2^3 * N, and so on.
After 55 minutes, the container is full. At that point, the number of bacteria is 2^(55/2) * N, which we can simplify to 2^27 * N.
Now, to find out how long it takes to fill a quarter of the container, we need to determine the number of bacteria when the container is one-quarter full. Since the number of bacteria doubles every 2 minutes, when the container is one-quarter full, the number of bacteria will be 2^(27-2n) * N, where n is the number of minutes it took to fill a quarter.
The number of bacteria when the container is a quarter full is equal to 0.25 times the number of bacteria when the container is full. Therefore, we can set up the equation:
0.25 * 2^27 * N = 2^(27-2n) * N
Now, let's solve this equation for n, which will give us the number of minutes it took to fill a quarter of the container.
First, let's simplify:
0.25 * 2^27 = 2^(27-2n)
Now, divide both sides of the equation by 2^27 to isolate the exponential term:
0.25 = 2^(-2n)
Next, take the logarithm base 2 of both sides to eliminate the exponential term:
log2(0.25) = log2(2^(-2n))
Using the logarithmic property log base a (x^y) = y * log base a (x), we can rewrite the equation as:
log2(0.25) = -2n * log2(2)
Since log2(2) equals 1, the equation simplifies to:
log2(0.25) = -2n
Now, calculate the logarithm base 2 of 0.25:
log2(0.25) ≈ -2
Now, solve for n by dividing both sides of the equation by -2:
n ≈ -2 / -2
n ≈ 1
Therefore, it took approximately 1 minute to fill a quarter of the container.