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The problem describes an exponential growth scenario where a colony of bacteria doubles every 20 minutes. The question asks, at what time the Petri dish was 12.5% covered, given that at noon it was completely (100%) covered.

Let’s break down the solution:

### Step-by-Step Solution

1. **Exponential growth formula**:
   The amount of bacteria follows the general rule for exponential growth:
   \[
   P(t) = P_0 \times 2^{t/T}
   \]
   Where:
   - \( P(t) \) is the amount of bacteria at time \( t \),
   - \( P_0 \) is the initial amount of bacteria,
   - \( T \) is the doubling time (20 minutes here),
   - \( t \) is the time passed.

2. **Relation to noon**:
   At noon, the dish is 100% covered. So, if we reverse time, 20 minutes earlier it was half covered (50%), 40 minutes earlier it was a quarter (25%), and so on.

3. **Finding 12.5%**:
   Since 12.5% is one-eighth ( \( \frac{1}{8} \) ) of the total population, we need to find how many doubling times back would give us \( \frac{1}{8} \) of the total.

   Each doubling decreases the coverage by half, so:
   \[
   \frac{1}{2}^n = \frac{1}{8}
   \]
   Solving for \( n \):
   \[
   n = 3
   \]
   This means 3 doubling periods back.

4. **Time Calculation**:
   Since each doubling period is 20 minutes, the bacteria would have been at 12.5% coverage 3 periods (3 \times 20 = 60 minutes) before noon.

### Answer:
The Petri dish was 12.5% covered **at 11:00 AM**.

Would you like further details or have any questions? Here are 5 related questions:

1. What if the doubling time was 10 minutes instead? How would the result change?
2. How does the exponential growth model differ from linear growth?
3. Can you derive the formula for bacteria population growth over time?
4. If the bacteria were only 6.25% covered, at what time would that have been?
5. How do we handle problems involving exponential decay versus growth?

**Tip**: Always check how far back in time you are when working with exponential growth—this helps you understand the past behavior of the system.

A colony of bacteria living on a Petri dish under optimal conditions doubles in size every 20 minutes. At noon a certain day, the Petri dish is completely covered with bacteria. At what time was 12.5% of the plate covered by bacteria?

Solve an exponential growth problem where a colony of bacteria doubles every 20 minutes. Learn how to calculate the time at which 12.5% of a Petri dish was covered before it was fully covered at noon. Suitable for high school students in Grades 9-12.

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