Question
The function V (r) = 10πr^2 represents to volume of a cylinder that is 10 inches high with a radius of r. What is the correct interpretation for the function notation V (3) = 90π?
A cylinder of 90π is produced from a radius of 3.
The volume of the cylinder is 3 when the radius is 90π
A radius of 3 gives the value 90
The volume of the cylinder is 90πr when the radius is 3
A cylinder of 90π is produced from a radius of 3.
The volume of the cylinder is 3 when the radius is 90π
A radius of 3 gives the value 90
The volume of the cylinder is 90πr when the radius is 3
Answers
Answered by
GPT-4o mini
To interpret the function notation \( V(3) = 90\pi \), we need to understand the function \( V(r) = 10\pi r^2 \) in terms of its parameters.
1. The function \( V(r) \) represents the volume of a cylinder with a height of 10 inches and a radius of \( r \).
2. When we substitute \( r = 3 \) into the function, we calculate the volume:
\[
V(3) = 10\pi (3^2) = 10\pi (9) = 90\pi
\]
Therefore, \( V(3) = 90\pi \) means that a cylinder with a radius of 3 inches has a volume of \( 90\pi \) cubic inches.
Now let's analyze the provided interpretations:
1. **A cylinder of 90π is produced from a radius of 3.** (This statement is somewhat vague but aligns with the fact that a cylinder with radius 3 has volume \( 90\pi \).)
2. **The volume of the cylinder is 3 when the radius is 90π.** (This statement is incorrect; the volume \( V(r) \) is not defined for a radius of \( 90\pi \) in this context.)
3. **A radius of 3 gives the value 90.** (This statement is incorrect; it gives \( 90\pi \), not just 90.)
4. **The volume of the cylinder is 90πr when the radius is 3.** (This is incorrect; the volume is \( 90\pi \), not \( 90\pi r \).)
The best interpretation among the choices could be seen as:
**A cylinder with a radius of 3 produces a volume of \( 90\pi \).**
However, this specific wording is not provided. The first option comes closest, but needs to clarify that it refers to volume, not just "a cylinder". Therefore, none of the provided interpretations accurately reflects what \( V(3) = 90\pi \) means directly.
If you have to choose from the provided options, option 1 is the least incorrect, but ideally, a more precise interpretation would be:
- "A cylinder with a radius of 3 inches has a volume of \( 90\pi \) cubic inches."
1. The function \( V(r) \) represents the volume of a cylinder with a height of 10 inches and a radius of \( r \).
2. When we substitute \( r = 3 \) into the function, we calculate the volume:
\[
V(3) = 10\pi (3^2) = 10\pi (9) = 90\pi
\]
Therefore, \( V(3) = 90\pi \) means that a cylinder with a radius of 3 inches has a volume of \( 90\pi \) cubic inches.
Now let's analyze the provided interpretations:
1. **A cylinder of 90π is produced from a radius of 3.** (This statement is somewhat vague but aligns with the fact that a cylinder with radius 3 has volume \( 90\pi \).)
2. **The volume of the cylinder is 3 when the radius is 90π.** (This statement is incorrect; the volume \( V(r) \) is not defined for a radius of \( 90\pi \) in this context.)
3. **A radius of 3 gives the value 90.** (This statement is incorrect; it gives \( 90\pi \), not just 90.)
4. **The volume of the cylinder is 90πr when the radius is 3.** (This is incorrect; the volume is \( 90\pi \), not \( 90\pi r \).)
The best interpretation among the choices could be seen as:
**A cylinder with a radius of 3 produces a volume of \( 90\pi \).**
However, this specific wording is not provided. The first option comes closest, but needs to clarify that it refers to volume, not just "a cylinder". Therefore, none of the provided interpretations accurately reflects what \( V(3) = 90\pi \) means directly.
If you have to choose from the provided options, option 1 is the least incorrect, but ideally, a more precise interpretation would be:
- "A cylinder with a radius of 3 inches has a volume of \( 90\pi \) cubic inches."