Let's solve this problem step-by-step.
### Step 1: Understand the Problem
We are given:
1. Initial number of bacteria: 2.9 million.
2. Growth rate: 4.6% every hour.
3. Duration: 3 days.
4. We need to calculate the final number of bacteria after 3 days and round our answer to the nearest tenth of a million.
### Step 2: Convert Growth Rate to a Multiplier
A 4.6% growth rate can be expressed as a multiplier. Growth rate per hour = 1 + 0.046 = 1.046.
### Step 3: Calculate Total Number of Hours
Since the growth rate is per hour, we need to figure out the total number of hours in 3 days:
- Total hours = 3 days × 24 hours/day = 72 hours
### Step 4: Apply the Compound Growth Formula
To find the final amount of bacteria, we use the compound growth formula:
[tex]\[ \text{Final Amount} = \text{Initial Amount} \times (\text{Growth Multiplier})^{\text{Total Hours}} \][/tex]
Let's plug in our values:
[tex]\[ \text{Final Amount} = 2.9 \times (1.046)^{72} \][/tex]
### Step 5: Calculate the Final Amount
[tex]\[ \text{Final Amount} \approx 73.90592962765206 \text{ million} \][/tex]
### Step 6: Round the Final Amount
We need to round the final amount to the nearest tenth of a million:
[tex]\[ 73.90592962765206 \approx 73.9 \text{ million} \][/tex]
### Conclusion
After 3 days, the number of bacteria will be approximately 73.9 million when rounded to the nearest tenth of a million.
