Question

the population of butterflies in a zoo exhibit is modeled by the following function, where x represents the number of months.
$b(x) = 435(0.76)^x$
which statement best describes this relationship?
a. the population of butterflies is increasing at a rate of 24%.
b. the population of butterflies is decreasing at a rate of 24%.
c. the population of butterflies is increasing at a rate of 76%.
d. the population of butterflies is decreasing at a rate of 76%.

Explanation:

Step1: Recall exponential decay formula

The general form of an exponential function is \( B(x)=a(b)^x \), where if \( 0 < b < 1 \), it represents exponential decay (decreasing population), and the decay rate \( r \) is given by \( r = 1 - b \).

Step2: Identify \( a \) and \( b \) in the given function

In the function \( B(x)=435(0.76)^x \), we have \( a = 435 \) and \( b = 0.76 \).

Step3: Calculate the decay rate

Since \( b = 0.76 \), the decay rate \( r=1 - 0.76 = 0.24 \), which is \( 24\% \). And since \( b = 0.76<1 \), the population is decreasing.

Answer:

B. The population of butterflies is decreasing at a rate of 24%.