Question

1. assignment: exponential growth and decay

the function $f(t) = 525(1.076)^t$ can be used to estimate the population of birds on an island after $t$ years.
which of the following statements are true?
select two correct answers.

• the initial population of birds decreased at a rate of 7.6% each year.
• the initial population of birds on the island is 1,076 birds.
• the growth factor of the population of birds on the island is 1.076.
• the population of birds after the first year is 525.
• the initial population of birds increased at a rate of 7.6% each year.

Explanation:

1. Analyze the exponential function \( f(t) = 525(1.076)^t \):

• The general form of exponential growth is \( f(t)=a(1 + r)^t \), where \( a \) is the initial amount, \( r \) is the growth rate, and \( 1 + r \) is the growth factor.
• For \( f(t) = 525(1.076)^t \), \( a = 525 \) (initial population), \( 1 + r=1.076 \), so \( r = 0.076=7.6\% \) (growth rate, not decay).
• Check each statement:
• "The initial population of birds decreased at a rate of 7.6% each year": False, since \( 1.076>1 \), it's growth, not decay.
• "The initial population of birds on the island is 1,076 birds": False, initial population \( a = 525 \).
• "The growth factor of the population of birds on the island is 1.076": True, as \( 1 + r = 1.076 \).
• "The population of birds after the first year is 525": False, at \( t = 1 \), \( f(1)=525(1.076)=525 + 525\times0.076

eq525 \).

• "The initial population of birds increased at a rate of 7.6% each year": True, \( r = 7.6\% \) and \( a = 525 \) (initial population) with growth.

1. Identify the two true statements.

Answer:

• The growth factor of the population of birds on the island is 1.076.
• The initial population of birds increased at a rate of 7.6% each year.