A hemispherical tank is full of water. It is connected to a pipe that empties it at a rate of 7 litres per second. How much approximate time (in minutes) will it take to empty half the tank if the internal radius of the tank is 2.1 meters? (Use π = $\frac{22}{7}$ and 1 \(m^{3}\)=1000 liters)
23 minutes
The tank is a hemisphere of internal radius \(r=2.1\ m\), and its volume is \(\dfrac{2}{3}\pi r^3\).
First cube the radius: \((2.1)^3=9.261\).
Full volume \(=\dfrac{2}{3}\times\dfrac{22}{7}\times9.261=19.404\ m^{3}\).
Half the tank is \(\dfrac{19.404}{2}=9.702\ m^{3}\), and converting with \(1\ m^{3}=1000\) litres gives \(9702\) litres.
The pipe empties \(7\) litres each second, so the time is \(\dfrac{9702}{7}=1386\) seconds.
Convert to minutes: \(\dfrac{1386}{60}=23.1\approx23\). The key steps are the hemisphere-volume formula and the \(m^{3}\)-to-litre conversion.
The time needed to empty half the tank is approximately 23 minutes.
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Important Questions from 3-D Mensuration
- Find the surface area of a sphere whose diameter is equal to 98 cm.
- A conical vessel has base radius 31 cm and height 45 cm. Water is poured into the vessel until it is $\frac{2}{3}$ full. Find the volume (in cm³) of water in the vessel.
- A solid metallic sphere of radius 10 cm is melted and recast into 125 identical spheres. What is the ratio of the surface area of the original sphere to the total surface area of 6 smaller spheres so formed?
- Find the volume (in cm³) of the largest right circular cone that can be cut out from a cube with an edge of 8 cm. Use $\pi = \frac{22}{7}$
- The volume of a solid cylinder is 5852 cm³ and its height is 38 cm. What is the total surface area of the solid cylinder? (Round your answer to the nearest integer) (Use $\pi = \frac{22}{7}$)
