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Sagot :
To solve the problem of determining the number of bacteria present at various times, we need to use the given growth function:
[tex]\[ f(t) = 500 \cdot e^{0.1t} \][/tex]
where [tex]\( f(t) \)[/tex] is the number of bacteria in millions, and [tex]\( t \)[/tex] is the time in days. Let's evaluate this function at three specific points: 3 days, 4 days, and 1 week (7 days).
### Step-by-Step Solution
#### (a) Number of bacteria present at 3 days
1. Substitute [tex]\( t = 3 \)[/tex] into the function:
[tex]\[ f(3) = 500 \cdot e^{0.1 \cdot 3} \][/tex]
2. Calculate the exponent:
[tex]\[ 0.1 \cdot 3 = 0.3 \][/tex]
3. Compute the value:
[tex]\[ e^{0.3} \approx 1.34986 \][/tex] (using a calculator or exponential table)
4. Multiply by 500:
[tex]\[ f(3) = 500 \cdot 1.34986 \approx 674.929 \][/tex]
So, the number of bacteria present at 3 days is approximately 674.929 million.
5. Round to the nearest integer:
[tex]\[ 674.929 \approx 675 \][/tex]
Therefore, approximately 675 million bacteria are present in 3 days.
#### (b) Number of bacteria present at 4 days
1. Substitute [tex]\( t = 4 \)[/tex] into the function:
[tex]\[ f(4) = 500 \cdot e^{0.1 \cdot 4} \][/tex]
2. Calculate the exponent:
[tex]\[ 0.1 \cdot 4 = 0.4 \][/tex]
3. Compute the value:
[tex]\[ e^{0.4} \approx 1.49182 \][/tex]
4. Multiply by 500:
[tex]\[ f(4) = 500 \cdot 1.49182 \approx 745.912 \][/tex]
So, the number of bacteria present at 4 days is approximately 745.912 million.
5. Round to the nearest integer:
[tex]\[ 745.912 \approx 746 \][/tex]
Therefore, approximately 746 million bacteria are present in 4 days.
#### (c) Number of bacteria present at 1 week (7 days)
1. Substitute [tex]\( t = 7 \)[/tex] into the function:
[tex]\[ f(7) = 500 \cdot e^{0.1 \cdot 7} \][/tex]
2. Calculate the exponent:
[tex]\[ 0.1 \cdot 7 = 0.7 \][/tex]
3. Compute the value:
[tex]\[ e^{0.7} \approx 2.01375 \][/tex]
4. Multiply by 500:
[tex]\[ f(7) = 500 \cdot 2.01375 \approx 1006.876 \][/tex]
So, the number of bacteria present at 7 days is approximately 1006.876 million.
5. Round to the nearest integer:
[tex]\[ 1006.876 \approx 1007 \][/tex]
Therefore, approximately 1007 million bacteria are present in 1 week.
### Conclusion
- (a) Approximately 675 million bacteria are present in 3 days.
- (b) Approximately 746 million bacteria are present in 4 days.
- (c) Approximately 1007 million bacteria are present in 1 week.
[tex]\[ f(t) = 500 \cdot e^{0.1t} \][/tex]
where [tex]\( f(t) \)[/tex] is the number of bacteria in millions, and [tex]\( t \)[/tex] is the time in days. Let's evaluate this function at three specific points: 3 days, 4 days, and 1 week (7 days).
### Step-by-Step Solution
#### (a) Number of bacteria present at 3 days
1. Substitute [tex]\( t = 3 \)[/tex] into the function:
[tex]\[ f(3) = 500 \cdot e^{0.1 \cdot 3} \][/tex]
2. Calculate the exponent:
[tex]\[ 0.1 \cdot 3 = 0.3 \][/tex]
3. Compute the value:
[tex]\[ e^{0.3} \approx 1.34986 \][/tex] (using a calculator or exponential table)
4. Multiply by 500:
[tex]\[ f(3) = 500 \cdot 1.34986 \approx 674.929 \][/tex]
So, the number of bacteria present at 3 days is approximately 674.929 million.
5. Round to the nearest integer:
[tex]\[ 674.929 \approx 675 \][/tex]
Therefore, approximately 675 million bacteria are present in 3 days.
#### (b) Number of bacteria present at 4 days
1. Substitute [tex]\( t = 4 \)[/tex] into the function:
[tex]\[ f(4) = 500 \cdot e^{0.1 \cdot 4} \][/tex]
2. Calculate the exponent:
[tex]\[ 0.1 \cdot 4 = 0.4 \][/tex]
3. Compute the value:
[tex]\[ e^{0.4} \approx 1.49182 \][/tex]
4. Multiply by 500:
[tex]\[ f(4) = 500 \cdot 1.49182 \approx 745.912 \][/tex]
So, the number of bacteria present at 4 days is approximately 745.912 million.
5. Round to the nearest integer:
[tex]\[ 745.912 \approx 746 \][/tex]
Therefore, approximately 746 million bacteria are present in 4 days.
#### (c) Number of bacteria present at 1 week (7 days)
1. Substitute [tex]\( t = 7 \)[/tex] into the function:
[tex]\[ f(7) = 500 \cdot e^{0.1 \cdot 7} \][/tex]
2. Calculate the exponent:
[tex]\[ 0.1 \cdot 7 = 0.7 \][/tex]
3. Compute the value:
[tex]\[ e^{0.7} \approx 2.01375 \][/tex]
4. Multiply by 500:
[tex]\[ f(7) = 500 \cdot 2.01375 \approx 1006.876 \][/tex]
So, the number of bacteria present at 7 days is approximately 1006.876 million.
5. Round to the nearest integer:
[tex]\[ 1006.876 \approx 1007 \][/tex]
Therefore, approximately 1007 million bacteria are present in 1 week.
### Conclusion
- (a) Approximately 675 million bacteria are present in 3 days.
- (b) Approximately 746 million bacteria are present in 4 days.
- (c) Approximately 1007 million bacteria are present in 1 week.
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