The formula r=Vπ−−√3

Question

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

What do the key features of the function reveal?

(1 point)
Responses

The function is an increasing function, so the length of the radius of the cylinder is increasing at an increasing rate.
The function is an increasing function, so the length of the radius of the cylinder is increasing at an increasing rate.

The domain is V≤0
, so only negative volumes are being studied.
The domain is cap v is less than or equal to 0 , so only negative volumes are being studied.

The function is an increasing function, so as the volume of the cylinder increases, the length of the radius of the cylinder increases but not 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 but not at a constant rate.

Due to the constraints, there is a minimum point at (0,0)
which indicates that when the volume of the cylinder is 0 cubic inches, the length of the radius of the cylinder is 10 inches.
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 cylinder is 0 cubic inches, the length of the radius of the cylinder is 10 inches.

Answer 1

To analyze the function \( r = \sqrt{\frac{V}{\pi}} \) in the context provided, we can break down the statements one by one:

1. The function is an increasing function, so the length of the radius of the cylinder is increasing at an increasing rate.

   • This statement is incorrect. While \( r \) does increase as \( V \) increases, the rate of increase of \( r \) is not increasing; it is actually decreasing because the square root function increases at a decreasing rate.

2. The domain is \( V \leq 0 \), so only negative volumes are being studied.

   • This statement is incorrect. The domain for the volume \( V \) of a cylinder should be \( V \geq 0 \) since negative volumes do not make sense in this context.

3. The function is an increasing function, so as the volume of the cylinder increases, the length of the radius of the cylinder increases but not at a constant rate.

   • This statement is correct. As \( V \) increases, \( r \) also increases, but the rate of increase diminishes as \( V \) grows larger because of the square root function.

4. Due to the constraints, there is a minimum point at (0,0), which indicates that when the volume of the cylinder is 0 cubic inches, the length of the radius of the cylinder is 10 inches.

   • This statement is incorrect. When \( V = 0 \), substituting in the equation gives \( r = \sqrt{\frac{0}{\pi}} = 0 \) inches, not 10 inches.

Based on this analysis, the most accurate statement about the features of the function is:

The function is an increasing function, so as the volume of the cylinder increases, the length of the radius of the cylinder increases but not at a constant rate.