After a certain medicine is ingested, the number of harmful bacteria remaining in the body declines rapidly.
The relationship between the elapsed time 
\[t\], in minutes, since the medicine is ingested, and the number of harmful bacteria remaining in the body, 
\[H_{\text{minute}}(t)\], is modeled by the following function:
\[H_{\text{minute}}(t)=500{,}000{,}000\cdot (0.2)^{t}\]
Complete the following sentence about the rate of change in the number of harmful bacteria remaining in the body in seconds.
Round your answer to two decimal places.
Every second, the number of harmful bacteria remaining in the body decays by a factor of 
.

The given model is \( H_{\text{minute}}(t) = 500,000,000 \cdot (0.2)^{t} \), where \( t \) is the time in minutes, and it models how the number of harmful bacteria decreases over time.

To determine the decay factor every second, we first need to understand the decay per minute and then adjust it to a per-second basis.

### Step 1: Decay factor per minute
The function \( H_{\text{minute}}(t) = 500,000,000 \cdot (0.2)^{t} \) shows that the bacteria decay by a factor of 0.2 per minute.

### Step 2: Converting to a decay factor per second
There are 60 seconds in one minute, so if \( t \) represents time in minutes, we want to determine the equivalent decay factor for each second.

Let \( H_{\text{second}}(t) \) represent the number of harmful bacteria as a function of time in seconds. To find the decay factor per second, we need to take the 60th root of the decay factor per minute. This is because the bacteria decay continuously, and each second will experience a smaller decay than the entire minute.

The decay factor per second is:
\[
(0.2)^{\frac{1}{60}}
\]

### Step 3: Calculating the decay factor per second
Now, let's compute this value.

The decay factor per second is approximately \( 0.97 \). 

Thus, every second, the number of harmful bacteria remaining in the body decays by a factor of **0.97**.

Would you like more details, or do you have any questions?

Here are 5 related questions to expand on this concept:
1. How long will it take for half of the bacteria to remain in the body?
2. What is the number of bacteria remaining after 10 minutes?
3. How does the decay factor change if the bacteria initially decayed by a factor of 0.1 per minute?
4. Can we express the bacteria decay as a continuous exponential decay model?
5. How does changing the initial number of bacteria affect the decay factor?

**Tip:** Exponential decay models are useful in various real-world contexts, including population studies and radioactive decay.

After a certain medicine is ingested, the number of harmful bacteria remaining in the body declines rapidly. The relationship between the elapsed time t, in minutes, since the medicine is ingested, and the number of harmful bacteria remaining in the body, H_minute(t), is modeled by the following function: H_minute(t)=500,000,000·(0.2)^t. Complete the following sentence about the rate of change in the number of harmful bacteria remaining in the body in seconds. Round your answer to two decimal places. Every second, the number of harmful bacteria remaining in the body decays by a factor of ____.

Learn how to calculate the decay factor per second using the exponential decay model. This problem involves determining the rate at which harmful bacteria decrease in the body after a medicine is ingested, modeled by H_minute(t)=500,000,000·(0.2)^t. Suitable for students in Grades 10-12.

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