Hourglass

An hourglass consists of two identical containers in the shape of rotational cones. For simplicity, we assume that the cones touch only at their apexes. The sand reaches to half the height of the lower cone. After turning the hourglass over, it takes exactly 21 minutes for the sand to pour from the upper container to the lower one. How many minutes would it take to pour all the sand if at the beginning the lower container was completely filled with sand?

Final Answer:

t₂ =  24 min

Step-by-step explanation:

$$\begin{gathered} t_1=21\text{min} \\ {t}_1\dots V_1 \\ {t}_2\dots V \\ {h}_1=h/2 \\ {V}_2=\frac{1}{3}\cdot \pi \cdot r_2^2\cdot (h/2) \\ V=\frac{1}{3}\cdot \pi \cdot r^2\cdot h \\ {r}_2:h_2=r:h \\ {r}_2:(h/2)=r:h \\ {r}_2=r/2 \\ {V}_2=\frac{1}{3}\cdot \pi \cdot (r/2{)}^2\cdot (h/2)=V/8 \\ {V}_2=V/8 \\ {V}_1=V-V_2=\frac{7}{8}V \\ {t}_2:V=t_1:V_1 \\ {t}_2=t_1\cdot V:V_1 \\ {t}_2=t_1\cdot V:(\frac{7}{8}V) \\ {t}_2=t_1\cdot \frac{8}{7}=21\cdot \frac{8}{7}=\frac{21\cdot 8}{7}=\frac{168}{7}=24\text{min} \end{gathered}$$

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