Question 1.4: 1.5 m³ of water has been poured in a conical tank of 2 m hei......
1.5 m³ of water has been poured in a conical tank of 2 m height and 1 m of base radius, as shown in Figure 1.5. Find the height of the free surface from the top of the cone.
Given,
\text { Volume of water, } V_{\text {water }}=1.5 m ^3
Height of conical tank, h = 2 m
Base radius, r = 1 m
\text { Total volume of conical flask }=\left\lgroup \frac{1}{3} \right\rgroup \pi r^2 h=\frac{1}{3} \times \pi \times 1 \times 2= 2.094 m³
Volume of water = 1.5 m³
Therefore, Additional water needed = 2.09 — 1.5 = 0.59 m³
Now, from Figure 1.5, considering the similarity of triangle OAB and OCD.
\begin{aligned} \frac{ OA }{ AB } & =\frac{ OC }{ CD } \\ \frac{h}{2} & =\frac{r_0}{1} \\ h & =2 r_0 \end{aligned}
Now,
\begin{aligned} \text { Volume of empty cone at top } & =\frac{\pi r_0^2 h}{3} \\ 0.59 & =\frac{\pi r_0^2 2 r_0}{3} \\ r_0 & =0.66 m \end{aligned}
Therefore, h = 2 × 0.66 = 1.32 m