Question

The equation V=43πr3 represents the relationship between the volume of a sphere and its radius. What does the end behavior tell you about the relationship between the volume of the sphere and its radius?(1 point) Responses As the radius decreases to negative infinity, the volume of the sphere will increase to infinity. As the radius decreases to negative infinity, the volume of the sphere will increase to infinity. As the radius increases to infinity, the volume of the sphere will increase to infinity. As the radius increases to infinity, the volume of the sphere will increase to infinity. As the radius decreases to negative infinity, the volume of the sphere will remain constant. As the radius decreases to negative infinity, the volume of the sphere will remain constant. As the radius increases to infinity, the volume of the sphere will decrease to negative infinity.

Answer 1

The equation \( V = 43\pi r^3 \) represents the volume of a sphere in terms of its radius \( r \). To analyze the end behavior and relationship between volume and radius, we can look at how \( V \) behaves as \( r \) changes.

1. As the radius \( r \) approaches positive infinity \( (r \to \infty) \):

   • Since \( V \) is directly proportional to \( r^3 \), as \( r \) increases, \( r^3 \) also increases without bound, which means that \( V \) will also increase without bound. Thus, the volume of the sphere increases to infinity.

2. As the radius \( r \) approaches negative infinity \( (r \to -\infty) \):

   • Since \( V = 43\pi r^3 \) and \( r^3 \) will be negative, \( V \) will also decrease without bound. Therefore, as \( r \) decreases to negative infinity, the volume will decrease towards negative infinity.

Given the possible responses, the correct answer is:

As the radius increases to infinity, the volume of the sphere will increase to infinity.