A tank has the shape of a cylinder with height 7m and base radius 3m. It is filled with water to a height of 5m. Find the work required to empty the tank by pumping all of the water to the top of the tank. (The density of water is 1000kg/m^3, and use g=9.8m/s^2.)
the volume of water is πr^2 h = 45π m^3
the weight of water is volume * density = 45000π kg * 9.8 = 1,386,856 N
The distance to raise the center of mass is 7-(5/2) = 4.5 m
work = force * distance = 6,240,852 J
or, if you insist on using calculus, you can go the long way, considering the work to raise a thin slice of water, which has thickness dy and is y m from the bottom of the tank:
∫[0,5] π*3^2*9.8*1000 * (7-y) dy
LOL do it the first way :)
To find the work required to empty the tank, we first need to calculate the volume of water in the tank.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
In this case, the base radius (r) is given as 3m and the water height (h) is 5m.
So, the volume of water in the tank is V = π(3^2)(5) = 45π m^3.
Next, we need to calculate the mass of the water in the tank. The density of water is given as 1000 kg/m^3, so the mass (m) of the water is:
m = density x volume = 1000 kg/m^3 x 45π m^3
Now, we can calculate the weight of the water in the tank. Weight (W) is given by the formula W = mass x gravity, where gravity (g) is 9.8 m/s^2.
So, W = m x g = (1000 kg/m^3 x 45π m^3) x 9.8 m/s^2.
Finally, to find the work required to empty the tank, we need to multiply the weight of the water by the height of the tank (7m).
Work (W) = Weight x height = (1000 kg/m^3 x 45π m^3 x 9.8 m/s^2) x 7m.
Simplifying this expression will give you the final answer.