Question
which statement best describes an informal argument for the formula for the volume of a cylinder?
A: a cylinder is a stack of r circles, each with an area of pi r^2 and a thickness of one unit, so its volume formula is V= pir^3
B: a cylinder is a curved lateral surface with area 2pi rh and two circular bases each with area pi r^2, so its volume formula is V=2 pi rh+2 pi r^2
C: a cylinder is a stack of h circles, each with an area of pi r^2 and a thickness of one unit, so its volume formula is V= pi r^2 h.]
D: a cylinder is a curved lateral surface with height h and two circular bases, each with area pi r^2, so its volume formula is V=2 pi r^2 h.
A: a cylinder is a stack of r circles, each with an area of pi r^2 and a thickness of one unit, so its volume formula is V= pir^3
B: a cylinder is a curved lateral surface with area 2pi rh and two circular bases each with area pi r^2, so its volume formula is V=2 pi rh+2 pi r^2
C: a cylinder is a stack of h circles, each with an area of pi r^2 and a thickness of one unit, so its volume formula is V= pi r^2 h.]
D: a cylinder is a curved lateral surface with height h and two circular bases, each with area pi r^2, so its volume formula is V=2 pi r^2 h.
Answers
GPT-4o mini
The statement that best describes an informal argument for the formula for the volume of a cylinder is:
**C: a cylinder is a stack of h circles, each with an area of pi r^2 and a thickness of one unit, so its volume formula is V= pi r^2 h.**
This explanation captures the essence of how the volume of a cylinder might be thought of informally as stacking circles (which are the bases of the cylinder) over the height of the cylinder, leading to the formula \( V = \pi r^2 h \).
**C: a cylinder is a stack of h circles, each with an area of pi r^2 and a thickness of one unit, so its volume formula is V= pi r^2 h.**
This explanation captures the essence of how the volume of a cylinder might be thought of informally as stacking circles (which are the bases of the cylinder) over the height of the cylinder, leading to the formula \( V = \pi r^2 h \).
you sure
GPT-4o mini
Yes, I am confident in my response. Option C provides the most straightforward and intuitive explanation for the volume of a cylinder, by describing it as a stack of circles (the bases) with a specific area (πr²) and extending that area through the height (h) of the cylinder. This aligns well with the mathematical formula \( V = \pi r^2 h \).
To clarify the other options briefly:
- **A:** Incorrect because it states the volume as \( V = \pi r^3 \), which is not the correct formula for the volume of a cylinder.
- **B:** Incorrect because it combines the lateral surface area and the base areas incorrectly for volume; volume should not include just the lateral surface area.
- **D:** Incorrect for the same reason as B; it misrepresents how volume is calculated by including lateral area, and also suggests an incorrect formula.
So, option C remains the best informal argument for deriving the volume of a cylinder.
To clarify the other options briefly:
- **A:** Incorrect because it states the volume as \( V = \pi r^3 \), which is not the correct formula for the volume of a cylinder.
- **B:** Incorrect because it combines the lateral surface area and the base areas incorrectly for volume; volume should not include just the lateral surface area.
- **D:** Incorrect for the same reason as B; it misrepresents how volume is calculated by including lateral area, and also suggests an incorrect formula.
So, option C remains the best informal argument for deriving the volume of a cylinder.