# Perimeter of base

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

Correct result:

D =  7.4 cm
h =  23.249 cm

#### Solution:

$V=1 \ l \rightarrow cm^3=1 \cdot \ 1000 \ cm^3=1000 \ cm^3 \ \\ o=h \ \\ 2 \pi \cdot \ r=h \ \\ \ \\ V=\pi \cdot \ r^2 \cdot \ h \ \\ V=\pi \cdot \ r^2 \cdot \ (2 \pi \cdot \ r) \ \\ V=2 \pi^2 \cdot \ r^3 \ \\ \ \\ r=\sqrt[3]{ V/ (2 \pi^2 )}=\sqrt[3]{ 1000/ (2 \cdot \ 3.1416^2 )} \doteq 3.7002 \ \text{cm} \ \\ \ \\ D=2 \cdot \ r=2 \cdot \ 3.7002=7.4 \ \text{cm}$
$h=2 \pi \cdot \ r=2 \cdot \ 3.1416 \cdot \ 3.7002 \doteq 23.2489 \doteq 23.249 \ \text{cm} \ \\ V_{2}=\pi \cdot \ r^2 \cdot \ h=3.1416 \cdot \ 3.7002^2 \cdot \ 23.2489=1000 \ \text{cm}^3 \ \\ V_{2}=V$

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