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Sagot :
To solve for the number of species in a sample using the given equation [tex]\( S(n) = a \ln \left( 1 + \frac{n}{a} \right) \)[/tex] where [tex]\(a = 0.48\)[/tex], let's calculate [tex]\( S(n) \)[/tex] for the specified values of [tex]\( n \)[/tex] step by step.
### Step-by-Step Calculations
#### Find [tex]\( S(100) \)[/tex]
1. Substitute [tex]\( n = 100 \)[/tex] and [tex]\( a = 0.48 \)[/tex] into the equation:
[tex]\[ S(100) = 0.48 \ln \left( 1 + \frac{100}{0.48} \right) \][/tex]
2. Calculate [tex]\( \frac{100}{0.48} \)[/tex]:
[tex]\[ \frac{100}{0.48} \approx 208.33 \][/tex]
3. Add 1 to the result:
[tex]\[ 1 + 208.33 \approx 209.33 \][/tex]
4. Calculate the natural logarithm of 209.33:
[tex]\[ \ln(209.33) \approx 5.34 \][/tex]
5. Multiply by the constant [tex]\( 0.48 \)[/tex]:
[tex]\[ 0.48 \times 5.34 \approx 2.56 \][/tex]
6. Round to the nearest whole number:
[tex]\[ S(100) \approx 3 \][/tex]
#### Find [tex]\( S(200) \)[/tex]
1. Substitute [tex]\( n = 200 \)[/tex] and [tex]\( a = 0.48 \)[/tex] into the equation:
[tex]\[ S(200) = 0.48 \ln \left( 1 + \frac{200}{0.48} \right) \][/tex]
2. Calculate [tex]\( \frac{200}{0.48} \)[/tex]:
[tex]\[ \frac{200}{0.48} \approx 416.67 \][/tex]
3. Add 1 to the result:
[tex]\[ 1 + 416.67 \approx 417.67 \][/tex]
4. Calculate the natural logarithm of 417.67:
[tex]\[ \ln(417.67) \approx 6.03 \][/tex]
5. Multiply by the constant [tex]\( 0.48 \)[/tex]:
[tex]\[ 0.48 \times 6.03 \approx 2.89 \][/tex]
6. Round to the nearest whole number:
[tex]\[ S(200) \approx 3 \][/tex]
#### Find [tex]\( S(150) \)[/tex]
1. Substitute [tex]\( n = 150 \)[/tex] and [tex]\( a = 0.48 \)[/tex] into the equation:
[tex]\[ S(150) = 0.48 \ln \left( 1 + \frac{150}{0.48} \right) \][/tex]
2. Calculate [tex]\( \frac{150}{0.48} \)[/tex]:
[tex]\[ \frac{150}{0.48} \approx 312.5 \][/tex]
3. Add 1 to the result:
[tex]\[ 1 + 312.5 \approx 313.5 \][/tex]
4. Calculate the natural logarithm of 313.5:
[tex]\[ \ln(313.5) \approx 5.75 \][/tex]
5. Multiply by the constant [tex]\( 0.48 \)[/tex]:
[tex]\[ 0.48 \times 5.75 \approx 2.76 \][/tex]
6. Round to the nearest whole number:
[tex]\[ S(150) \approx 3 \][/tex]
#### Find [tex]\( S(10) \)[/tex]
1. Substitute [tex]\( n = 10 \)[/tex] and [tex]\( a = 0.48 \)[/tex] into the equation:
[tex]\[ S(10) = 0.48 \ln \left( 1 + \frac{10}{0.48} \right) \][/tex]
2. Calculate [tex]\( \frac{10}{0.48} \)[/tex]:
[tex]\[ \frac{10}{0.48} \approx 20.83 \][/tex]
3. Add 1 to the result:
[tex]\[ 1 + 20.83 \approx 21.83 \][/tex]
4. Calculate the natural logarithm of 21.83:
[tex]\[ \ln(21.83) \approx 3.08 \][/tex]
5. Multiply by the constant [tex]\( 0.48 \)[/tex]:
[tex]\[ 0.48 \times 3.08 \approx 1.48 \][/tex]
6. Round to the nearest whole number:
[tex]\[ S(10) \approx 1 \][/tex]
### Final Results
- [tex]\( S(100) = 3 \)[/tex]
- [tex]\( S(200) = 3 \)[/tex]
- [tex]\( S(150) = 3 \)[/tex]
- [tex]\( S(10) = 1 \)[/tex]
### Step-by-Step Calculations
#### Find [tex]\( S(100) \)[/tex]
1. Substitute [tex]\( n = 100 \)[/tex] and [tex]\( a = 0.48 \)[/tex] into the equation:
[tex]\[ S(100) = 0.48 \ln \left( 1 + \frac{100}{0.48} \right) \][/tex]
2. Calculate [tex]\( \frac{100}{0.48} \)[/tex]:
[tex]\[ \frac{100}{0.48} \approx 208.33 \][/tex]
3. Add 1 to the result:
[tex]\[ 1 + 208.33 \approx 209.33 \][/tex]
4. Calculate the natural logarithm of 209.33:
[tex]\[ \ln(209.33) \approx 5.34 \][/tex]
5. Multiply by the constant [tex]\( 0.48 \)[/tex]:
[tex]\[ 0.48 \times 5.34 \approx 2.56 \][/tex]
6. Round to the nearest whole number:
[tex]\[ S(100) \approx 3 \][/tex]
#### Find [tex]\( S(200) \)[/tex]
1. Substitute [tex]\( n = 200 \)[/tex] and [tex]\( a = 0.48 \)[/tex] into the equation:
[tex]\[ S(200) = 0.48 \ln \left( 1 + \frac{200}{0.48} \right) \][/tex]
2. Calculate [tex]\( \frac{200}{0.48} \)[/tex]:
[tex]\[ \frac{200}{0.48} \approx 416.67 \][/tex]
3. Add 1 to the result:
[tex]\[ 1 + 416.67 \approx 417.67 \][/tex]
4. Calculate the natural logarithm of 417.67:
[tex]\[ \ln(417.67) \approx 6.03 \][/tex]
5. Multiply by the constant [tex]\( 0.48 \)[/tex]:
[tex]\[ 0.48 \times 6.03 \approx 2.89 \][/tex]
6. Round to the nearest whole number:
[tex]\[ S(200) \approx 3 \][/tex]
#### Find [tex]\( S(150) \)[/tex]
1. Substitute [tex]\( n = 150 \)[/tex] and [tex]\( a = 0.48 \)[/tex] into the equation:
[tex]\[ S(150) = 0.48 \ln \left( 1 + \frac{150}{0.48} \right) \][/tex]
2. Calculate [tex]\( \frac{150}{0.48} \)[/tex]:
[tex]\[ \frac{150}{0.48} \approx 312.5 \][/tex]
3. Add 1 to the result:
[tex]\[ 1 + 312.5 \approx 313.5 \][/tex]
4. Calculate the natural logarithm of 313.5:
[tex]\[ \ln(313.5) \approx 5.75 \][/tex]
5. Multiply by the constant [tex]\( 0.48 \)[/tex]:
[tex]\[ 0.48 \times 5.75 \approx 2.76 \][/tex]
6. Round to the nearest whole number:
[tex]\[ S(150) \approx 3 \][/tex]
#### Find [tex]\( S(10) \)[/tex]
1. Substitute [tex]\( n = 10 \)[/tex] and [tex]\( a = 0.48 \)[/tex] into the equation:
[tex]\[ S(10) = 0.48 \ln \left( 1 + \frac{10}{0.48} \right) \][/tex]
2. Calculate [tex]\( \frac{10}{0.48} \)[/tex]:
[tex]\[ \frac{10}{0.48} \approx 20.83 \][/tex]
3. Add 1 to the result:
[tex]\[ 1 + 20.83 \approx 21.83 \][/tex]
4. Calculate the natural logarithm of 21.83:
[tex]\[ \ln(21.83) \approx 3.08 \][/tex]
5. Multiply by the constant [tex]\( 0.48 \)[/tex]:
[tex]\[ 0.48 \times 3.08 \approx 1.48 \][/tex]
6. Round to the nearest whole number:
[tex]\[ S(10) \approx 1 \][/tex]
### Final Results
- [tex]\( S(100) = 3 \)[/tex]
- [tex]\( S(200) = 3 \)[/tex]
- [tex]\( S(150) = 3 \)[/tex]
- [tex]\( S(10) = 1 \)[/tex]
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