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 smaller than the initial butterfly population of field A, but the population of both fields is increasing at the same rate.
C.
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.
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 population of butterflies in field B is smaller than in field A, with B(0) = 25 and A(0) = 50. However, both populations grow at the same rate since both have a growth factor of 1.728 per year. Therefore, statement B is correct.

Step-by-step explanation:

A scientist observed the population of butterflies in two different fields. In field A, the population is defined by the function A(x) = 50(1.2)^{3x}, and in field B, the population is defined by the function B(x) = 25(1.728)^x. To determine which statement is true, we must evaluate the initial populations and the rate of increase for each field.

For field A, the initial population when x equals 0 is A(0) = 50(1.2)^0 = 50. For field B, the initial population when x equals 0 is B(0) = 25(1.728)^0 = 25. Therefore, the initial population of field B is indeed smaller than that of field A.

Next, we examine the growth rates. In field A, the expression 1.2^3, or 1.728, represents the growth factor per year. In field B, the growth factor per year is also 1.728. This indicates that both fields have the exact same growth rate.

Thus, the correct statement is B: 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.