The growth of a certain species​ (in millions) since

1950
 
closely fits the following exponential function where t is the number of years since

1950

.

Upper A left parenthesis t right parenthesis equals 3500 e Superscript 0.0166 t

a. The population of the species was about

4142
 
million in

1960

. 

How closely does the function approximate this​ value?

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

8137
 
​million)

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

Given the exponential growth function for the species population:

\[
A(t) = 3500 e^{0.0166t}
\]

where \( t \) is the number of years since 1950.

### Part (a): Approximating the population in 1960
In 1960, \( t = 1960 - 1950 = 10 \) years.

We calculate the population using the given function:

\[
A(10) = 3500 e^{0.0166 \times 10}
\]

Now, let's compute that value and compare it with the actual population in 1960, which was about 4142 million.

### Part (b): Approximating the population in 2000
In 2000, \( t = 2000 - 1950 = 50 \) years.

We calculate the population using the function:

\[
A(50) = 3500 e^{0.0166 \times 50}
\]

This result will be compared with the actual population in 2000, which was about 8137 million.

### Part (c): Estimating the population in 2015
In 2015, \( t = 2015 - 1950 = 65 \) years.

We calculate the population using the function:

\[
A(65) = 3500 e^{0.0166 \times 65}
\]

Let me compute all of these values for you.### Results:
- **(a)** The calculated population in 1960 using the function is approximately \( 4132 \) million, while the actual population was 4142 million. The function provides a close approximation, with a difference of about 10 million.
  
- **(b)** The calculated population in 2000 is approximately \( 8027 \) million, while the actual population was 8137 million. The difference is about 110 million, indicating that the function slightly underestimates the population in 2000.

- **(c)** The estimated population in 2015 using the function is approximately \( 10296 \) million.

Would you like more details or have any questions?

Here are 5 follow-up questions related to this topic:
1. How does changing the growth rate affect population predictions over time?
2. Can you derive the doubling time of the species' population using the given growth rate?
3. How would the population change if the initial population were 4000 million instead of 3500 million?
4. What is the relative error between the predicted and actual populations in 1960 and 2000?
5. How can this exponential model be adapted if the growth rate is expected to change over time?

**Tip:** In exponential growth, small changes in the growth rate can have a significant impact over long time periods.

The growth of a certain species (in millions) since 1950 closely fits the following exponential function where t is the number of years since 1950. A(t) = 3500 e^0.0166t. 

 a. The population of the species was about 4142 million in 1960. How closely does the function approximate this value?
 b. Use the function to approximate the population of the species in 2000. (The actual population in 2000 was about 8137 million)
 c. Estimate the population of the species in the year 2015.

Explore how the population of a species since 1950 is modeled using the exponential growth function A(t) = 3500 e^0.0166t. Learn to approximate populations in 1960, 2000, and 2015 using this model, and compare them with actual data. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation