A hemispherical tank, full of water, is emptied by a pipe at the rate of 22/7 litres per second. How much time it will take to empty half the tank if the diameter of the base of the tank is 3m?
the volume of a sphere of diameter d is π/6 d^3
so, you want to empty 1/4 of the sphere (half a hemisphere), or
π/24 * 3^3 = 9π/8 m^3 = 9000π/8 L
At a rate of π L/s, that will take 9000/8 = 1125 seconds
or 18.75 minutes
The given answer of this question is 16 min 30 sec
To find the time it will take to empty half the tank, we need to determine the volume of half the tank and divide it by the rate at which water is being emptied.
First, we need to find the volume of the hemispherical tank. The formula for the volume of a hemisphere is (2/3) * π * r^3, where r is the radius of the base.
Given that the diameter of the base of the tank is 3m, the radius (r) can be calculated by dividing the diameter by 2:
r = 3m / 2 = 1.5m
Now we can calculate the volume of the hemispherical tank:
Volume = (2/3) * π * (1.5m)^3
Volume = (2/3) * 22/7 * 1.5m * 1.5m * 1.5m
Volume = (2/3) * 22/7 * 3.375m^3
Volume = (2/3) * (22/7) * 3.375m^3
Volume ≈ 14.25m^3
Therefore, the volume of half the tank is:
Volume of half the tank = 1/2 * 14.25m^3
Volume of half the tank ≈ 7.125m^3
Next, we need to determine the time it takes to empty this volume at a rate of 22/7 liters per second. To convert the volume from cubic meters to liters, we need to multiply by 1000 (since 1 cubic meter = 1000 liters).
Volume of half the tank in liters = 7.125m^3 * 1000
Volume of half the tank in liters ≈ 7125 liters
Now we can find the time it will take to empty half the tank:
Time = Volume of half the tank / Rate of emptying
Time = 7125 liters / (22/7 liters per second)
Time = 7125 liters / (22/7) liters/second
To divide by a fraction, we multiply by its reciprocal:
Time = 7125 liters * (7/22) seconds/liter
Time = 7125 * 7 / 22 seconds
Time ≈ 2271.59 seconds
Therefore, it will take approximately 2271.59 seconds to empty half the tank.