Three cylinders with volumes $V_{1}, V_{2}$, and $V_{3}$ have heights and base radii equal to 10 and $10 \mathrm{~cm}, 10$ and $5 \mathrm{~cm}$, and 20 and $5 \mathrm{~cm}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_30c9a294a58a6e4b8190g-016.jpg?height=345&width=400&top_left_y=256&top_left_x=862)

(a) Write in ascending order the volumes $V_{1}, V_{2}$, and $V_{3}$ of the three cylinders.

(b) Give the dimensions of a cylinder whose volume $V_{4}$ is between $V_{2}$ and $V_{3}$.

(c) Give the dimensions of a cylinder whose volume $V_{5}$ is between $V_{1}$ and $V_{3}$.
The volume of a cylinder with radius $R$ and height $h$ is given by $\pi R^{2} h$.

(a) The three volumes are $V_{1}=\pi \times 10^{3}=1000 \pi, V_{2}=\pi \times 5^{2} \times 10=250 \pi$ and $V_{3}=$ $\pi \times 5^{2} \times 20=500 \pi$, therefore, $V_{1}>V_{3}>V_{2}$.

(b) Since cylinders $V_{2}$ and $V_{3}$ have the same radius, it is sufficient to keep the radius of the cylinder at $5 \text{ cm}$ and the height between 10 and $20 \text{ cm}$; for example, $h=15 \text{ cm}$. In this case, the volume $V_{4}$ of the new cylinder is $\pi \times 5^{2} \times 15=375 \pi \text{ cm}^{3}$.
(c) To construct a cylinder with volume $V_{5}$ between $V_{1}$ and $V_{3}$, we can take the smaller of the two heights, which is $10 \text{ cm}$, and reduce the radius of the larger volume cylinder from 10 to $8 \text{ cm}$, obtaining a cylinder with volume $V_{5}=\pi \times 8^{2} \times 10=640 \pi \text{ cm}^{3}$.