Question

Water is pouring into a conical vessel as shown in the adjoining figure at the rate of 1.8 cubic meter per minute. How long will it take to fill the vessel?

Answer 1

Hint: For solving this problem, first calculate the total volume of the conical vessel by using the given diameter and slant height. The rate of water flowing into the vessel is 1.8 cubic metre per minute. By dividing the total volume of the vessel with the rate, we get the time taken to fill the vessel.

Complete step-by-step solution -
According to the problem statement, we are given the diameter and slant height of a conical vessel as 5.2 metre (radius = 2.6m) and 6.8 metre respectively. We find the perpendicular height of the conical vessel by using Pythagoras theorem which can be stated as: ${{H}^{2}}={{P}^{2}}+{{B}^{2}}$
$\begin{align}
 \Rightarrow & {{l}^{2}}={{r}^{2}}+{{h}^{2}} \\
  \Rightarrow & {{6.8}^{2}}={{2.6}^{2}}+{{h}^{2}} \\
 \Rightarrow & {{h}^{2}}=46.24-6.76 \\
 \Rightarrow & {{h}^{2}}=39.48 \\
 \Rightarrow & h=\sqrt{39.48} \\
 \Rightarrow & h=6.283m \\
\end{align}$

Therefore, the height of the conical vessel is 6.283 m.
The volume formula for a cone is $V=\dfrac{{1}}{{3}} \pi {{r}^{2}}h$.
By using the volume formula and putting r = 2.6m and h = 6.283m, we get the volume of conical vessel as
$\begin{align}
  \Rightarrow & V=\dfrac{\pi }{3}\times 2.6\times 2.6\times 6.283 \\
 \Rightarrow & V=\dfrac{22}{7\times 3}\times 2.6\times 2.6\times 6.283 \\
 \Rightarrow & V=44.4776{{m}^{3}} \\
\end{align}$
Rate of flow of water is 1.8 cubic metre per minute.
On dividing the volume of a conical vessel with the rate we get the time taken to fill the vessel.
$\begin{align}
  & \text{time = }\dfrac{\text{volume}}{rate} \\
  \Rightarrow & t=\dfrac{44.4776{{m}^{3}}}{1.8{{m}^{3}}/\min } \\
 \Rightarrow & t=24.7098\min \\
 \Rightarrow & t\approx 25\min \\
\end{align}$
Therefore, the time taken to fill the vessel is 25 minutes.

Note: Students must be careful while calculating the time taken to fill the vessel because the rate is given in minutes. So, the final time will be obtained in minutes. The volume of cones should be remembered for solving this problem.