Question

the population of a species in 2025 was 4 billion and the relative growth rate was estimated at 15 percent per year. assuming that the population follows an exponential growth model, find the projected population in 2034. round your answer to 1 decimal place. the population will be billion in 2034

Explanation:

Step1: Recall the exponential growth formula

The exponential growth model is given by \( P(t) = P_0 e^{rt} \), where \( P_0 \) is the initial population, \( r \) is the relative growth rate (in decimal), and \( t \) is the time in years.

Step2: Determine the values of \( P_0 \), \( r \), and \( t \)

• \( P_0 = 4 \) billion (population in 2025)
• \( r = 0.15 \) (15% per year, converted to decimal)
• \( t = 2034 - 2025 = 9 \) years (time from 2025 to 2034)

Step3: Substitute the values into the formula

Substitute \( P_0 = 4 \), \( r = 0.15 \), and \( t = 9 \) into \( P(t) = P_0 e^{rt} \):
\( P(9) = 4 \times e^{0.15 \times 9} \)

Step4: Calculate the exponent

First, calculate \( 0.15 \times 9 = 1.35 \)

Step5: Calculate \( e^{1.35} \)

Using a calculator, \( e^{1.35} \approx 3.857425534 \)

Step6: Calculate the population

Multiply by \( P_0 = 4 \):
\( P(9) = 4 \times 3.857425534 \approx 15.42970214 \)

Step7: Round to 1 decimal place

Rounding \( 15.42970214 \) to 1 decimal place gives \( 15.4 \)

Answer:

15.4