Let's analyze the problem step by step.
### Part a: Approximation for the population in 1970
The given function \( A(t) \) is used to approximate the species population in millions since 1960.
In 1970, \( t = 10 \) years since 1960.
Using the given function to calculate the population for \( t = 10 \):
[tex]\[ \text{population\_1970} \approx 1484.67 \text{ million} \][/tex]
The actual population in 1970 was 4142 million.
Next, we find the difference:
[tex]\[ \text{difference\_1970} = 4142 - 1484.67 = 2657.33 \text{ million} \][/tex]
To determine how significant this difference is, we compute the percentage difference:
[tex]\[ \text{percentage\_difference\_1970} = \left( \frac{2657.33}{4142} \right) \times 100 \approx 64.16\% \][/tex]
Since the difference and percentage difference are quite substantial, the calculated value is not close to the actual value of 4142 million.
Thus, the correct choice is:
A. The population of the species in 1970 is approximately 1484.67 million. This is not a close approximation to the actual value of 4142 million, because the difference in the two populations is large, both in absolute and percentage terms.
### Part b: Approximation for the population in 2000
In 2000, \( t = 40 \) years since 1960.
Using the given function to calculate the population for \( t = 40 \):
[tex]\[ \text{population\_2000} \approx 1081593 \text{ million} \][/tex]
Rounding to the nearest whole number:
[tex]\[ \text{approximate population in 2000} = 1081593 \text{ million} \][/tex]
### Part c: Estimation for the population in 2015
In 2015, \( t = 55 \) years since 1960.
Using the given function to calculate the population for \( t = 55 \):
[tex]\[ \text{population\_2015} \approx 29193174 \text{ million} \][/tex]
The step-by-step solution shows how the function is used to approximate and compare the species population at given times, leading to the following results:
1. For 1970:
- Population from the function: approximately 1484.67 million
- Difference from actual value: 2657.33 million
- Percentage difference: ~64.16%
- Conclusion: Not a close approximation; Large difference
2. For 2000:
- Approximate population: 1081593 million (rounded to the nearest whole number)
3. For 2015:
- Estimated population: 29193174 million
