Final answer:
The work required to pump all the water out of the hemispherical tank is approximately 0.8305 MJ.
Step-by-step explanation:
The work W done to pump all the water out of the tank can be calculated using the equation:
W = mgh
where:
- m is the mass of the water,
- g is the acceleration due to gravity,
- h is the height through which the water is lifted.
Given that the tank is a hemisphere, the volume V of water can be calculated using the formula for the volume of a hemisphere:
V = {2} / {3}π r^3
where r is the radius of the hemisphere.
The mass m of the water is then the product of the volume and the density ρ of water:
m = ρ V
Now, substitute V from the hemisphere volume formula into the mass formula:
m = ρ ({2} / {3}π r^3)
Now, substitute (m), (g), and (h) into the work formula:
W = ρ (2 / 3 π r^3)gh
Given the values:
- ( ρ = 1000 kg/m^3 ) (density of water),
- ( g = 9.8 m/s^2 ) (acceleration due to gravity),
- ( r = 8 m ) (radius of the hemisphere),
- ( h = 1 m) (height through which water is lifted),
Substitute these values into the formula to calculate the work W.
W = 1000 (2 / 3 π (8)^3)(9.8)(1)
W ≈ 8.305 x 10^5 J
Now, convert the work from joules to megajoules:
W ≈ 0.8305 MJ
Therefore, the work required to pump all the water out of the tank is approximately 0.8305 MJ.