A tank in the shape of a right circular cylinder is filled with water (62.5 lb/ft3). It has a height of 8 ft and a diameter of 10 ft. How much work is required to pump all the water to a spout that is 3 ft above the top of the tank?
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To find the work required to pump all the water out of the tank, we need to calculate the weight of the water and then find the work done in lifting it to the spout.
Step 1: Calculate the volume of the tank
The tank is in the shape of a right circular cylinder. The formula to calculate the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.
Given that the diameter is 10 ft, the radius (r) is half of that, so r = 10/2 = 5 ft.
The height (h) of the tank is 8 ft.
So, the volume (V) of the tank is V = π(5^2)(8) = 200π ft^3.
Step 2: Calculate the weight of water
The weight of water is given as 62.5 lb/ft^3. So, the weight of the water in the tank is the volume of the tank multiplied by the density.
Weight = Volume × Density = 200π ft^3 × 62.5 lb/ft^3.
Step 3: Calculate the work done in lifting the water
The work done (W) is equal to the weight of the water multiplied by the distance it is lifted.
Given that the distance to lift the water is 3 ft above the top of the tank, the work done is W = Weight × Distance = (200π ft^3 × 62.5 lb/ft^3) × 3 ft.
Step 4: Evaluate the expression
Now, we can calculate the work required by evaluating the expression W = (200π ft^3 × 62.5 lb/ft^3) × 3 ft.
W = (200π × 62.5 × 3) lb·ft.
Calculating this expression will give you the work required to pump all the water to the spout that is 3 ft above the top of the tank.