A spherical tank with radius of 5 feet is set on a coloumn 15 feet above the ground. How much work is required to fill the tank with water if the solution is pumped from ground level?
Let the center of the tank be at (0,0)
when the water level is at y, the radius of the water surface is
r^2 = 25-y^2
Work is ∫F(y) dy
F(y) is the weight of the water. So, since each slice of water in the tank is raised 21+y feet, and water weighs 62.4 lbs/ft^3,
W(y) = ∫[-5,5] 62.4 π (25-y^2) (21+y) dy
= 218400π
To calculate the work required to fill the tank, we first need to determine the volume of the tank. Since it is a spherical tank, we can find the volume using the formula:
Volume of a sphere = (4/3) * π * r^3
Given that the radius of the tank is 5 feet, we can substitute this value into the formula:
Volume = (4/3) * π * 5^3
= (4/3) * π * 125
= 500π cubic feet
Next, we need to determine the height water needs to be pumped to fill the tank. The height is equal to the distance from the ground level to the bottom of the tank, which is the height of the column plus the radius of the tank:
Height to pump = Height of column + Radius of tank
= 15 feet + 5 feet
= 20 feet
Now, we have enough information to calculate the work required using the formula:
Work = Weight * Height
The weight of the water can be calculated using the density of water, which is 62.4 pounds per cubic foot. Therefore, the weight is equal to the volume of water (which we found earlier) multiplied by the density:
Weight = Volume * Density
= 500π * 62.4 pounds
Finally, we can substitute the weight and the height into the formula to calculate the work:
Work = Weight * Height
= (500π * 62.4) * 20
= 624,000π foot-pounds
So, the work required to fill the spherical tank with water is approximately 624,000π foot-pounds.