# Cylinder horizontally

The cylinder with a diameter of 3 m and a height/length of 15 m is laid horizontally. Water is poured into it, reaching a height of 60 cm below the axis of the cylinder. How many hectoliters of water is in the cylinder?

Correct result:

V =  267.527 hl

#### Solution:

$D=3 \ \text{m} \ \\ R=D/2=3/2=\dfrac{ 3 }{ 2 }=1.5 \ \text{m} \ \\ L=15 \ \text{m} \ \\ r=60/100=\dfrac{ 3 }{ 5 }=0.6 \ \text{m} \ \\ h=R - r=1.5 - 0.6=\dfrac{ 9 }{ 10 }=0.9 \ \text{m} \ \\ S_{1}=R^2 \cdot \ \arccos(r/R)=1.5^2 \cdot \ \arccos(0.6/1.5) \doteq 2.6084 \ \text{m}^2 \ \\ S_{2}=r \cdot \ \sqrt{ 2 \cdot \ R \cdot \ h-h^2 }=0.6 \cdot \ \sqrt{ 2 \cdot \ 1.5 \cdot \ 0.9-0.9^2 } \doteq 0.8249 \ \text{m}^2 \ \\ S=S_{1} - S_{2}=2.6084 - 0.8249 \doteq 1.7835 \ \text{m}^2 \ \\ V_{1}=L \cdot \ S=15 \cdot \ 1.7835 \doteq 26.7527 \ \text{m}^3 \ \\ V=V_{1} \rightarrow hl=V_{1} \cdot \ 10 \ hl=26.7527280982 \cdot \ 10 \ hl=267.527 \ hl=267.527 \ \text{hl}$

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