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

Answer 1

The equation given, \( V = 43\pi r^3 \), represents the volume \( V \) of a sphere in terms of its radius \( r \).

To analyze the end behavior of this equation, we can consider what happens as \( r \) increases or decreases:

1. As \( r \) increases to infinity (i.e., \( r \to \infty \)):

   • The term \( r^3 \) will also increase to infinity.
   • Thus, \( V \) will increase to infinity as well since it is directly proportional to \( r^3 \).

2. As \( r \) decreases to negative infinity (i.e., \( r \to -\infty \)):

   • The term \( r^3 \) will become a large negative number (since the cube of a negative number is negative).
   • Consequently, \( V \) will also decrease to negative infinity.

3. As \( r \) decreases to 0 (i.e., \( r \to 0 \)):

   • The volume will decrease to 0 since \( V = 43 \pi (0)^3 = 0 \).

Given this analysis, the correct statement about the end behavior of the relationship between the volume of the sphere and its radius is:

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