An initial number of bacteria presented in a culture is 10000. This number doubles every 30 minutes. How long will it take to bacteria to reach the number 100000 ?

Explanation

To solve the problem of how long it will take for the bacteria to grow from 10,000 to 100,000, we can follow these steps:

### Step 1: Understand the Growth of Bacteria
The initial number of bacteria is given as 10,000. The bacteria double every 30 minutes.

### Step 2: Set Up the Equation
Let \( t \) be the time in minutes. The number of bacteria after \( t \) minutes can be expressed as:
\[ N(t) = 10,000 \times 2^{(t/30)} \]
where \( N(t) \) is the number of bacteria at time \( t \).

### Step 3: Set the Target Number of Bacteria
We want to find out when the number of bacteria reaches 100,000:
\[ 100,000 = 10,000 \times 2^{(t/30)} \]

### Step 4: Simplify the Equation
Divide both sides by 10,000:
\[ 10 = 2^{(t/30)} \]

### Step 5: Take Logarithm of Both Sides
To solve for \( t \), take the logarithm (base 2) of both sides:
\[ \log_2(10) = \frac{t}{30} \]

### Step 6: Solve for \( t \)
Now, multiply both sides by 30 to isolate \( t \):
\[ t = 30 \times \log_2(10) \]

### Step 7: Calculate \( \log_2(10) \)
Using the change of base formula:
\[ \log_2(10) = \frac{\log_{10}(10)}{\log_{10}(2)} = \frac{1}{\log_{10}(2)} \approx \frac{1}{0.301} \approx 3.32193 \]

### Step 8: Substitute Back to Find \( t \)
Now plug this value back into the equation for \( t \):
\[ t \approx 30 \times 3.32193 \approx 99.6579 \]

### Step 9: Round to the Nearest Whole Number
Rounding this to the nearest whole number, we get:
\[ t \approx 100 \text{ minutes} \]

### Final Answer
It will take approximately **100 minutes** for the bacteria to grow from 10,000 to 100,000.

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