Consider a trough with triangular ends where the tank is 10 feet long, top is 5 feet wide, and the tank is 4 feet deep. Say that the trough is full to within 1 foot of the top with water of weight density 62.4 pounds/ft^3, and pump is used to empty the tank until the water remaining in the tank is 1 foot deep. Find the total work to accomplish the task.
at height y from the bottom, the area of the surface of the water is (5/4)y*10 ft^2
So, the weight of a thin layer of water at that height is (5/4)y*10*dy *62.4 = 780y dy lbs
So, the work to lift all the sheets of water at height y is
∫[1,3] 780y(4-y) dy = 5720 ft-lbs
To find the total work required to empty the tank, we need to calculate the weight of the water and then find the work done in lifting this weight.
1. First, let's calculate the volume of the water in the tank. Since the tank has triangular ends, we can assume it is a prism with a triangular base.
The triangular base has a width of 5 feet and a height of 4 feet (depth of the tank). Therefore, the area of the triangular base is (1/2) * 5 * 4 = 10 square feet.
The length of the tank is given as 10 feet, so the volume of water in the tank is given by:
Volume = base area * length = 10 * 10 = 100 cubic feet.
2. Next, we can calculate the weight of the water using the weight density provided.
Weight = density * volume = 62.4 * 100 = 6240 pounds.
3. Now that we have the weight of the water, we can calculate the work done in lifting this weight.
The work done in lifting an object is given by the formula:
Work = Force * Distance
In this case, the force is equal to the weight of the water, which is 6240 pounds.
The distance is the vertical distance over which the water is lifted. The water level needs to be lowered from a depth of 3 feet (1 foot from the top) to a depth of 1 foot. Therefore, the vertical distance is 3 - 1 = 2 feet.
Work = Force * Distance = 6240 * 2 = 12,480 foot-pounds.
So, the total work required to empty the tank is 12,480 foot-pounds.
To find the total work required to empty the tank, we can use the principle of work done against gravity. The work done against gravity is equal to the force exerted by the water multiplied by the distance it is lifted.
Step 1: Calculate the weight of the water in the tank
The weight of the water can be calculated by multiplying the volume of water by its weight density.
Volume of water = length × width × depth
V = 10 ft × 5 ft × ((4 ft - 1 ft) + (4 ft - 1 ft))/2 (since the trough has triangular ends)
Step 2: Convert the volume to cubic feet
The depth of the water in the trough is given in feet, so we don't need any conversions here.
Step 3: Calculate the weight of the water
Weight of water = Volume × Weight density
W = V × 62.4 lbs/ft³
Step 4: Calculate the work done against gravity
The work done against gravity is given by the formula:
Work = Weight × Distance lifted
In this case, the distance lifted is the depth of the water that remains in the tank after pumping.
Work = W × (4 ft - 1 ft) = W × 3 ft
Therefore, the total work required to accomplish the task is 3 times the weight of the water in the tank.
Note: The calculations provided above assume that there are no losses due to friction or inefficiencies in the pumping process.