The formula r=3Vπ−−−√3

Question

can be used to find the length of the radius of a cone in feet where the height is equal to the radius for a cone with a volume of V cubic feet.

What do the key features of the function reveal?

(1 point)
Responses

The range is r≤0
, so the radius of a cone only has negative values.
The range is r is less than or equal to 0 , so the radius of a cone only has negative values.

The function is an increasing function, so as the volume of the cylinder increases, the length of the radius of the cylinder increases at a constant rate.
The function is an increasing function, so as the volume of the cylinder increases, the length of the radius of the cylinder increases at a constant rate.

Due to the constraints, there is a minimum point at (0,0)
which indicates that when the volume of the cone is 10 cubic feet, the length of the radius of the cone is 0 feet.
Due to the constraints, there is a minimum point at open paren 0 comma 0 close paren which indicates that when the volume of the cone is 10 cubic feet, the length of the radius of the cone is 0 feet.

The function is an increasing function, so the length of the radius of the cone is increasing at a decreasing rate.
The function is an increasing function, so the length of the radius of the cone is increasing at a decreasing rate.

Answer 1

To analyze the function \( r = \sqrt[3]{\frac{3V}{\pi}} \), we can derive some key features from the formula.

1. Understanding the Function:

   • The formula expresses the relationship between the volume \( V \) of a cone and its radius \( r \).
   • Since \( V \) is a volume, it is non-negative, meaning \( V \geq 0 \).

2. Range of the Function:

   • The formula indicates that \( r \) (the radius) is determined by the cube root of a non-negative number (since \( \frac{3V}{\pi} \geq 0 \) for \( V \geq 0 \)).
   • Thus, \( r \) can be 0 or positive, meaning \( r \geq 0 \).

3. Behavior of the Function:

   • As \( V \) increases, \( r \) will also increase since the cube root function is an increasing function. This means if you increase \( V \), \( r \) will increase.
   • However, the rate of increase of \( r \) will not be constant; it will increase at a decreasing rate due to the nature of the cube root function.

4. Key Points:

   • At \( V = 0 \), \( r = 0 \), indicating that if there is no volume, there is no radius.
   • The function is not defined for negative volumes since volume cannot be negative in this context.

Given this analysis, the most accurate interpretation of the features of the function is:

The function is an increasing function, so the length of the radius of the cone is increasing at a decreasing rate.

This reflects that as the volume increases, the radius increases, but the rate of increase diminishes.