A bacterium divides every `35` minutes. If a culture containing `10^(5)` cells/ml is grown for `175` minutes. What will be the cell concentration / ml after `175` minutes

Options

A. `175xx10^(5)`
B. `35xx10^(5)`
C. `5xx10^(5)`
D. `32xx10^(5)`

Answer

To solve the problem, we need to determine the cell concentration of a bacterium after a certain period of time, given its division rate.

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Initial cell concentration = \(10^5\) cells/ml
- Division time = 35 minutes
- Total time for growth = 175 minutes

2. **Calculate the Number of Generations:**
- To find out how many times the bacteria will divide in 175 minutes, we divide the total time by the division time:
\[ \text{Number of generations} = \frac{\text{Total time}}{\text{Division time}} = \frac{175 \text{ minutes}}{35 \text{ minutes}} = 5 \]

3. **Determine the Growth Factor:**
- Since the bacteria double in number with each generation, the growth factor after \(n\) generations is given by \(2^n\).
- For 5 generations, the growth factor will be:
\[ \text{Growth factor} = 2^5 = 32 \]

4. **Calculate the Final Cell Concentration:**
- The final concentration of cells after 175 minutes can be calculated by multiplying the initial concentration by the growth factor:
\[ \text{Final concentration} = \text{Initial concentration} \times \text{Growth factor} = 10^5 \text{ cells/ml} \times 32 \]
- Therefore, the final concentration is:
\[ \text{Final concentration} = 32 \times 10^5 \text{ cells/ml} \]

5. **Conclusion:**
- After 175 minutes, the cell concentration will be \(32 \times 10^5\) cells/ml.

### Final Answer:
The cell concentration after 175 minutes will be \(32 \times 10^5\) cells/ml.

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