Question

A scientist observed the population of butterflies in two different fields. In field A, the population of butterflies is defined by the function A(x) = 50(1.2)^(3x), where x represents the number of years. In field B, the population of butterflies is defined by the function B(x) = 25(1.728)^x, where x represents the number of years. Which statement is true?

A) The initial butterfly population of field B is smaller than the initial butterfly population of field A, but the population of both fields is decreasing at the same rate.
B) The initial butterfly population of field B is greater than the initial butterfly population of field A, but the population of both fields is decreasing at the same rate.
C) The initial butterfly population of field B is smaller than the initial butterfly population of field A, but the population of both fields is increasing at the same rate.
D) The initial butterfly population of field B is greater than the initial butterfly population of field A, but the population of both fields is increasing at the same rate.

Answer 1

The initial butterfly population of field B is smaller than that of field A, but both populations are increasing at the same rate of 1.728 per year, making statement C the correct answer.

Step-by-step explanation:

The question asks us to compare the initial populations and growth rates of butterflies in two different fields, A and B, based on the provided exponential functions: A(x) = 50(1.2)^(3x) for field A and B(x) = 25(1.728)^x for field B.

The initial butterfly population of each field can be found by evaluating the function at x=0. For A(0), we get A(0) = 50(1.2)^0 = 50, and for B(0), we get B(0) = 25(1.728)^0 = 25. Since A(0) > B(0), the initial population of field A is greater than field B.

Next, we examine the growth rates by looking at the base of the exponents. In A(x), the base is 1.2 raised to the power of 3x, meaning the growth factor per year is 1.2^3, or 1.728. In B(x), the base is 1.728 raised to the power of x, which is the same growth factor as in A(x) per year. Hence, both populations are increasing at the same rate.

The correct statement is therefore: C) The initial butterfly population of field B is smaller than the initial butterfly population of field A, but the population of both fields is increasing at the same rate.