Conical bottle

When a conical bottle rests on its flat base, the water in the bottle is 8 cm from its 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?

Correct answer:

h =  10.2195 cm

Step-by-step explanation:

$$\begin{gathered} h_1=8\doteq 10.2195\text{cm} \\ {h}_2=2\doteq -8.2195\text{cm} \\ V=\frac{1}{3}\pi r^{2}h \\ {V}_3=\frac{1}{3}\pi r_1^2{h}_1 \\ {V}_1=V-V_3 \\ {V}_1=\frac{1}{3}\pi r^{2}h-\frac{1}{3}\pi r_1^2{h}_1 \\ {V}_1=\frac{1}{3}\pi (r^{2}h-r_1^2{h}_1) \\ {r}_1<r_2<r \\ {r}_1:h_1=r:h \\ {r}_1=r\cdot h_1/h \\ {r}_2:(h-h_2)=r:h \\ {r}_2=r\cdot (h-h_2)/h \\ {V}_2=\frac{1}{3}\pi r_2^2(h-h_2) \\ {V}_1=V_2 \\ \frac{1}{3}\pi (r^{2}h-r_1^2{h}_1)=\frac{1}{3}\pi r_2^2(h-h_2) \\ (r^{2}h-r_1^2{h}_1)=r_2^2(h-h_2) \\ (r^{2}h-(r\cdot h_1/h{)}^2{h}_1)=(r(h-h_2)/h{)}^2(h-h_2) \\ (h-(h_1/h{)}^2\cdot h_1)=((h-h_2)/h{)}^2(h-h_2) \\ (h-(8/h{)}^2\cdot 8)=((h-2)/h{)}^2(h-2) \\ h(h-2)=84 \\ {h}^2-2h-84=0 \\ a=1;b=-2;c=-84 \\ D=b^2-4ac=2^2-4\cdot 1\cdot (-84)=340 \\ D>0 \\ {h}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{2\pm \sqrt{340}}{2}=\frac{2\pm 2\sqrt{85}}{2} \\ {h}_{1,2}=1\pm 9.219544 \\ {h}_1=10.219544457 \\ {h}_2=-8.219544457 \\ h>0 \\ h=h_1=10.2195=10.2195\text{cm} \end{gathered}$$

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