Answer:
[tex]\begin{array}{ccc}\text{Radius}&\text{Volume of sphere}&\text{Volume of cylinder}\\&&\\1&\dfrac{4}{3}\pi &2\pi \\&&\\2&\dfrac{32}{3}\pi &16\pi \\&&\\3&36\pi &54\pi \\&&\\4&\dfrac{256}{3}\pi &128\pi \\&&\\5&\dfrac{500}{3}\pi &250\pi\end{array}[/tex]
Step-by-step explanation:
Use formulas for the volumes:
[tex]V_{sphere}=\dfrac{4}{3}\pi r^3,\\ \\V_{cylinder}=\pi r^2h=\pi r^2\cdot 2r=2\pi r^3.[/tex]
1. When r=1,
[tex]V_{sphere}=\dfrac{4}{3}\pi\cdot 1^3=\dfrac{4}{3}\pi,\\ \\V_{cylinder}=2\pi \cdot 1^3=2\pi.[/tex]
2. When r=2,
[tex]V_{sphere}=\dfrac{4}{3}\pi\cdot 2^3=\dfrac{32}{3}\pi,\\ \\V_{cylinder}=2\pi \cdot 2^3=16\pi.[/tex]
3. When r=3,
[tex]V_{sphere}=\dfrac{4}{3}\pi\cdot 3^3=36\pi,\\ \\V_{cylinder}=2\pi \cdot 3^3=54\pi.[/tex]
4. When r=4,
[tex]V_{sphere}=\dfrac{4}{3}\pi\cdot 4^3=\dfrac{256}{3}\pi,\\ \\V_{cylinder}=2\pi \cdot 4^3=128\pi.[/tex]
5. When r=5,
[tex]V_{sphere}=\dfrac{4}{3}\pi\cdot 5^3=\dfrac{500}{3}\pi,\\ \\V_{cylinder}=2\pi \cdot 5^3=250\pi.[/tex]
Note that for all r,
[tex]\dfrac{V_{sphere}}{V_{cylinder}}=\dfrac{\frac{4}{3}\pi r^3}{2\pi r^3}=\dfrac{2}{3}.[/tex]
