Given a starting population of 100 bacteria, the formula `b(t)=100(2^(t))` can be used to determine the number of bacteria, b, after t periods of time. If each time period is 15 minutes long, how many minutes will it take for the population of bacteria to reach 51,200?

Answer

To solve the problem of how long it will take for a population of bacteria to reach 51,200 given the starting population and growth formula, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Growth Formula**: The formula given for the bacteria population is:
\[ b(t) = 100 \times 2^t \]
where \( b(t) \) is the number of bacteria after \( t \) time periods.

2. **Set Up the Equation**: We need to find \( t \) when the population reaches 51,200. Therefore, we set up the equation:
\[ 51,200 = 100 \times 2^t \]

3. **Divide Both Sides by 100**: To isolate \( 2^t \), divide both sides of the equation by 100:
\[ \frac{51,200}{100} = 2^t \]
Simplifying the left side gives:
\[ 512 = 2^t \]

4. **Express 512 as a Power of 2**: We need to express 512 as a power of 2. We know that:
\[ 512 = 2^9 \]
Therefore, we can rewrite the equation as:
\[ 2^9 = 2^t \]

5. **Equate the Exponents**: Since the bases are the same, we can equate the exponents:
\[ t = 9 \]

6. **Calculate the Total Time in Minutes**: Each time period is 15 minutes long. To find the total time in minutes, multiply the number of time periods by the duration of each period:
\[ \text{Total time} = 15 \times t = 15 \times 9 = 135 \text{ minutes} \]

### Final Answer:
It will take **135 minutes** for the population of bacteria to reach 51,200.

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