Perimeter of base

The circumference of the base of the rotating cylinder is the same as its height. What is the diameter and height of this cylinder with a volume of 1 liter?

Final Answer:

D =  7.4004 cm
h =  23.2489 cm

Step-by-step explanation:

$$\begin{gathered} V=1\text{l}\to {\text{cm}}^3=1\cdot 1000{\text{cm}}^3=1000{\text{cm}}^3 \\ o=h \\ 2\cdot \pi \cdot r=h \\ V=\pi \cdot r^2\cdot h \\ V=\pi \cdot r^2\cdot (2\pi \cdot r) \\ V=2\cdot {\pi }^2\cdot r^3 \\ r=\sqrt[3]{V/(2{\pi }^2)}=\sqrt[3]{1000/(2\cdot 3.1416^2)}\doteq 3.7002\text{cm} \\ D=2\cdot r=2\cdot 3.7002=7.4004\text{cm} \end{gathered}$$

$$\begin{gathered} h=2\pi \cdot r=2\cdot 3.1416\cdot 3.7002\doteq 23.2489\text{cm} \\ {V}_2=\pi \cdot r^2\cdot h=3.1416\cdot 3.7002^2\cdot 23.2489=1000{\text{cm}}^3 \\ {V}_2=V \end{gathered}$$

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