It was observed in an experiment that the number of bacteria doubles evey hour. It was found that the number of bacteria twelve hours from the start of observation was 40960. After how many hours from the start of the experiment would the number of bacteria has been one-fourth the final number of bacteria?

Options

A. 11
B. 10
C. 8
D. 6

Answer

To solve the problem step by step, we will use the information given about the bacteria doubling every hour and the final count after 12 hours.

### Step 1: Understand the growth of bacteria
The number of bacteria doubles every hour. This means if we denote the initial number of bacteria as \( A \), after \( n \) hours, the number of bacteria can be expressed as:
\[ N(n) = A \cdot 2^n \]

### Step 2: Use the information given
We know that after 12 hours, the number of bacteria is 40,960. Therefore, we can set up the equation:
\[ N(12) = A \cdot 2^{12} = 40,960 \]

### Step 3: Solve for \( A \)
To find \( A \), we first calculate \( 2^{12} \):
\[ 2^{12} = 4096 \]
Now we can substitute this back into our equation:
\[ A \cdot 4096 = 40,960 \]
To find \( A \), we divide both sides by 4096:
\[ A = \frac{40,960}{4096} = 10 \]

### Step 4: Find the final number of bacteria
Now we know the initial number of bacteria \( A \) is 10. The final number of bacteria after 12 hours is:
\[ N(12) = 10 \cdot 2^{12} = 40,960 \]

### Step 5: Determine when the bacteria count is one-fourth of the final count
We need to find the time \( t \) when the number of bacteria is one-fourth of the final number:
\[ N(t) = \frac{1}{4} \cdot N(12) = \frac{1}{4} \cdot 40,960 = 10,240 \]
Using the formula for \( N(t) \):
\[ N(t) = A \cdot 2^t = 10 \cdot 2^t \]
Setting this equal to 10,240:
\[ 10 \cdot 2^t = 10,240 \]
Dividing both sides by 10:
\[ 2^t = 1,024 \]

### Step 6: Solve for \( t \)
Now we need to express 1,024 as a power of 2:
\[ 1,024 = 2^{10} \]
Thus, we have:
\[ 2^t = 2^{10} \]
This implies:
\[ t = 10 \]

### Conclusion
The number of bacteria will be one-fourth of the final number after **10 hours** from the start of the experiment.