a cattle trough has a trapezoidal cross section with a height of 1 m and horizontal sides of width 1/2 m and 1 m assume the length of the trough is 10 m. A. how much work is required to pump out the water in the trough? B. if the length is doubled does the amount of work needed double? explain.

The trapezoidal cross-section for water at depth 1/2 + h/2 = (1+h)/2

So, the surface area of the water at depth h is 10*(1+h)/2 = 5(1+h)

So, the volume of the sheet of water is

5(1+h) dh

That means that the weight of the water (at 1000kg/m^3) layer is

5(1+h) dh * 1000 * 9.8

Since work = force (weight) * distance, the work required to lift the layer of water to the top of the trough is

5(1+h) dh * 1000 * 9.8 * (1-h)
= 49000 (1-h^2) dh

Now just integrate that from 0 to y where y is the depth of the water, and you have your answer.

Note that since the length of the trough is a constant, doubling the length just doubles the work.

To find the amount of work required to pump out the water in the trough, we need to calculate the volume of water in the trough and then multiply it by the weight of the water.

A. Calculating the volume of water:
The cross-sectional area of the trough changes along its length due to its trapezoidal shape. We can divide the trough into smaller sections to make calculations easier.

1. Calculate the cross-sectional area of the smaller section at the bottom of the trough:
The smaller section is a rectangle with a width of 0.5 m (1/2 m) and a height of 1 m. So, the cross-sectional area is 0.5 m * 1 m = 0.5 square meters.

2. Calculate the cross-sectional area of the larger section at the top of the trough:
The larger section is a trapezoid with a width of 1 m and a height of 1 m. So, the cross-sectional area is (0.5 m + 1 m) * 1 m / 2 = 0.75 square meters.

3. Calculate the average area of the smaller and larger sections:
The average area can be found by adding the cross-sectional areas from Step 1 and Step 2 and then dividing by 2:
Average area = (0.5 square meters + 0.75 square meters) / 2 = 0.625 square meters.

4. Calculate the volume of water in one meter length of the trough:
The volume of water is found by multiplying the average area by the length of the trough section:
Volume = 0.625 square meters * 1 meter = 0.625 cubic meters.

B. If the length of the trough is doubled:
Now, let's assume the length of the trough is doubled to 20 m and see if the amount of work needed to pump out the water also doubles.

1. Calculate the volume of water in the longer trough:
Using the same method as above, the volume of water in one meter length of the longer trough is still 0.625 cubic meters.

2. Calculate the volume of water in the entire 20 m length trough:
Since the volume is the same for each meter length, and the length is now 20 m, the total volume of water is 0.625 cubic meters * 20 meters = 12.5 cubic meters.

Conclusion:
A. The amount of work required to pump out the water in the trough can be found by multiplying the volume of water by the weight of the water.
B. If the length of the trough is doubled, the amount of work needed also doubles since the volume of water increases proportionally.

To calculate the work required to pump out the water in the trough, we need to find the volume of water in the trough and then multiply it by the weight of water.

A. Finding the volume of water in the trough:
1. Divide the trough into two parts: a rectangular prism and a triangular prism.
2. Calculate the volume of each part separately and then add them together.

Volume of the rectangular prism:
Volume_rectangular = base_area * height
= (width1 + width2) * height / 2 * height
= (1/2 + 1) * 1 / 2 * 1
= 3/4 m³

Volume of the triangular prism:
Volume_triangular = base_area * height / 2
= width1 * height / 2 * height
= 1/2 * 1 / 2 * 1
= 1/4 m³

Total volume of water in the trough:
Volume_total = Volume_rectangular + Volume_triangular
= 3/4 + 1/4
= 1 m³

B. Calculating the work required to pump out the water:
The work required to pump out the water is given by the formula:
Work = force × distance

In this case, force is the weight of the water, and distance is the height of the trough.

Weight of the water:
Weight = density × volume × acceleration due to gravity

Assuming the density of water is 1000 kg/m³ and acceleration due to gravity is 9.8 m/s², we can calculate the weight of the water:
Weight = 1000 * 1 * 9.8
= 9800 N

Work required to pump out the water:
Work = Weight * distance
= 9800 * 1
= 9800 J (Joules)

B. If the length is doubled, does the amount of work needed double?
No, the amount of work needed does not double if the length is doubled. The volume of water increases with the length, but the height and cross-sectional area remain the same. Therefore, the work required will increase, but it will not be directly proportional to the length.