Question
Josiah tried the problem above and decides that the best way to make them all the same volume is the keep them all with a radius of 2 and a height of 9. That way, each bottle is the same size and will hold the same volume.
Do you agree with Josiah, why or why not? Use mathematical reasoning to support your answer.
Do you agree with Josiah, why or why not? Use mathematical reasoning to support your answer.
Answers
GPT-4o mini
To determine if Josiah's approach of using bottles with a radius of 2 and a height of 9 will ensure that they all have the same volume, we need to calculate the volume of a cylinder using the formula:
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height of the cylinder.
In this case, with \( r = 2 \) and \( h = 9 \):
\[
V = \pi (2)^2 (9)
\]
Calculating this:
\[
V = \pi (4)(9) = 36\pi
\]
This means that each bottle with the specified dimensions will have a volume of \( 36\pi \).
If all bottles are made with these dimensions (radius 2 and height 9), then indeed, they will all have the same volume of \( 36\pi \).
Thus, I agree with Josiah because if all bottles are designed with the same radius and height, they will hold the same volume. Since volume is a function of both radius and height in the context of cylinders, uniform dimensions lead to uniform volume. Therefore, Josiah's decision to use a radius of 2 and a height of 9 is mathematically sound for ensuring that all bottles have the same volume.
\[
V = \pi r^2 h
\]
where \( r \) is the radius and \( h \) is the height of the cylinder.
In this case, with \( r = 2 \) and \( h = 9 \):
\[
V = \pi (2)^2 (9)
\]
Calculating this:
\[
V = \pi (4)(9) = 36\pi
\]
This means that each bottle with the specified dimensions will have a volume of \( 36\pi \).
If all bottles are made with these dimensions (radius 2 and height 9), then indeed, they will all have the same volume of \( 36\pi \).
Thus, I agree with Josiah because if all bottles are designed with the same radius and height, they will hold the same volume. Since volume is a function of both radius and height in the context of cylinders, uniform dimensions lead to uniform volume. Therefore, Josiah's decision to use a radius of 2 and a height of 9 is mathematically sound for ensuring that all bottles have the same volume.