# Conical bottle

When a conical bottle rests on its flat base, the water in the bottle is 8 cm from it vertex. When the same conical bottle is turned upside down, the water level is 2 cm from its base. What is the height of the bottle?

Result

h =  10.22 cm

#### Solution:

$h_{1}=8 \doteq 10.2195 \ \text{cm} \ \\ h_{2}=2 \doteq -8.2195 \ \text{cm} \ \\ \ \\ V=\dfrac{ 1 }{ 3 } \pi r^2 \ h \ \\ V_{3}=\dfrac{ 1 }{ 3 } \pi r_{1}^2 \ h_{1} \ \\ \ \\ V_{1}=V-V_{3} \ \\ V_{1}=\dfrac{ 1 }{ 3 } \pi r^2 \ h-\dfrac{ 1 }{ 3 } \pi r_{1}^2 \ h_{1} \ \\ V_{1}=\dfrac{ 1 }{ 3 } \pi (r^2 \ h - r_{1}^2 \ h_{1}) \ \\ \ \\ r_{1}0 \ \\ \ \\ h_{1,2}=\dfrac{ -b \pm \sqrt{ D } }{ 2a }=\dfrac{ 2 \pm \sqrt{ 340 } }{ 2 }=\dfrac{ 2 \pm 2 \sqrt{ 85 } }{ 2 } \ \\ h_{1,2}=1 \pm 9.2195444572929 \ \\ h_{1}=10.219544457293 \ \\ h_{2}=-8.2195444572929 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (h -10.219544457293) (h +8.2195444572929)=0 \ \\ \ \\ h>0 \ \\ \ \\ h=h_{1}=10.2195 \doteq 10.2195 \doteq 10.22 \ \text{cm}$

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