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The growth of a certain species (in millions) since 1960 closely fits the following exponential function where [tex] t [/tex] is the number of years since 1960.

[tex] A(t) = 350000166e^{0.02t} [/tex]

a. The population of the species was about 4142 million in 1970. How closely does the function approximate this value?

Choose the correct answer below:

A. The population of the species in 1970 is approximately 4132 million. This is not a close approximation to the actual value of 4142 million, because the difference in the two populations is ten million, which is large.

B. The population of the species in 1970 is approximately 4132 million. This is fairly close to the actual value of 4142 million, because the percentage difference is very small.

C. The population of the species in 1970 is approximately 3632 million. This is not a close approximation to the actual value of 4142 million, because the approximation does not take into account a massive growth increase between 1960 and 1970.

b. Use the function to approximate the population of the species in 2000. (The actual population in 2000 was about 6909 million.)

The approximate population in 2000 is __________ million. (Round to the nearest whole number as needed.)

c. Estimate the population of the species in the year 2015.



Answer :

GinnyAnswer
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