Question

Water is pouring into a conical vessel of diameter 5.2 m and slant height 6.8 m, at the rate of $1.8 m^3$ per minute. How long will it take to fill the vessel?

Answer 1

Hint: Water is flowing into the vessel at a constant rate. As the water flows into the vessel, the level of the water increases. When the vessel is completely full, the quantity of water inside the vessel is equal to the volume of the vessel. The rate to fill a vessel is given by the total volume added or removed per unit time. The volume of a cone is given by-
$\dfrac13\pi\times r^2\times\mathrm h$
Here r is the radius of the cone and h is its height.

Complete step-by-step answer:

We will first find the volume of the cone and then will divide it by the rate to find the time taken to fill the cone.
Diameter of the cone = 5.2 m
So, radius of the cone = 2.6 m
Slant height of the cone = 6.8 m
Using the relation ${{\text{h}}^2} + {{\text{r}}^2} = {{\text{l}}^2}$
Here h is the height of the cone, r is the radius and l is the slant height.
Substituting the given values of r and l we get,
$h^2 + (2.6)^2 = (6.8)^2$
$h^2 = 39.48$
h = 6.28 m
Using the formula, the volume of the cone is-
$=\dfrac13\times3.14\times2.6^2\times6.28\\$
$= 44.43 m^3$
Now, the rate of flow is given by-
$\mathrm r=\dfrac{\mathrm V}{\mathrm t}\\$
On cross multiplying we get-
V = rt
44.43 = 1.8t
t = 24.68 minutes.
The time taken to fill the vessel is 24.68 minutes. This is the required answer.

Note:Students often get confused in the units of the answer and write it in second. But as we can see the units of the rate is in $m^3/minute$, hence the time calculated will also be in minutes. Also, students often use the value of slant height in the formula for volume, which is completely incorrect and should be avoided