A trough has a trapezoidal cross section with a height of 5 m and horizontal sides of width 2.5 m and 5 m. Assume the length of the trough is 14 m.

How much work (in joules) is required to pump the water out of the trough (to the level of the top of the trough) when it is full?

If the length is doubled is the required work doubled?

Why did the water go to therapy? Because it had some trapezoidal issues!

To calculate the work required to pump the water out of the trough, we need to find the volume of water in the trough first. The volume of a trapezoidal trough can be calculated using the formula:

Volume = (1/2) × (b1 + b2) × h × L

where b1 and b2 are the lengths of the bottom and top bases, h is the height of the trough, and L is the length of the trough.

In this case, b1 = 2.5 m, b2 = 5 m, h = 5 m, and L = 14 m. Plugging these values into the formula, we get:

Volume = (1/2) × (2.5 + 5) × 5 × 14 = 8.75 m^3

Now, to find the work required, we need to multiply the volume by the weight of the water and the height it needs to be pumped. The weight of water is approximately 1000 kg/m^3, and as we need to pump it to the height of the trough (5 m), the work required can be calculated as:

Work = Volume × Weight × Height
= 8.75 m^3 × 1000 kg/m^3 × 5 m
= 43,750 joules (approximately)

So, approximately 43,750 joules of work is required to pump the water out of the trough when it's full.

Now let's double the length of the trough and see if the required work is doubled. If the length is doubled to 28 m, the volume of water will also double, but the other parameters remain the same. Therefore, the new volume will be 2 × 8.75 m^3 = 17.5 m^3.

Using the same formula as before, we can calculate the new work:

Work = 17.5 m^3 × 1000 kg/m^3 × 5 m
= 87,500 joules (approximately)

So, when the length of the trough is doubled, the required work is also doubled.

Remember, though the work might double, the laughs from my jokes remain constant!

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

1. Calculate the area of the trapezoidal cross section:
Area = (base1 + base2) * height / 2
The bases are 2.5 m and 5 m, and the height is 5 m.
Area = (2.5 + 5) * 5 / 2 = 18.75 square meters

2. Calculate the volume of water in the trough:
Volume = Area * length
The length of the trough is 14 m.
Volume = 18.75 * 14 = 262.5 cubic meters

3. Convert the volume of water to work:
Work = Volume * weight of water * height
The weight of water is approximately 1000 kg/m^3, and the height is 5 m.
Work = 262.5 * 1000 * 5 = 1,312,500 joules

Therefore, the amount of work required to pump the water out of the trough when it is full is 1,312,500 joules.

If the length is doubled, the required work is not necessarily doubled. The volume of water in the trough will be doubled, but the work required will depend on the height the water needs to be pumped and the efficiency of the pump.

To find the work required to pump the water out of the trough, we need to calculate the volume of water in the trough first. The volume of a trapezoidal trough can be determined by multiplying the average width of the cross-section by the length and the height.

Here's how to find the volume of water in the trough:
1. Find the average width of the cross-section: (2.5 + 5) / 2 = 3.75 meters
2. Multiply the average width by the length and the height: 3.75 * 14 * 5 = 262.5 cubic meters

Since the volume is in cubic meters, we need to convert it to cubic centimeters to match the unit of work (joules). We know that 1 cubic meter is equal to 1,000,000 cubic centimeters, so:

262.5 cubic meters * 1,000,000 cubic centimeters/cubic meter = 262,500,000 cubic centimeters

Now, to calculate the work required to pump the water out, we can use the equation:

Work = Force * Distance

The force required to pump the water out is equal to the weight of the water, which can be calculated using the density of water (1 gram/cubic centimeter) and the volume of water in cubic centimeters:

Weight = Volume * Density = 262,500,000 * 1 = 262,500,000 grams

To convert the weight to joules, we need to multiply it by the acceleration due to gravity (9.8 m/s²), and the conversion factor to convert grams to joules (1 gram * 1 cm²/1 joule)

Work = Weight * Gravity * Conversion Factor

Work = 262,500,000 grams * 9.8 m/s² * 1 joule/gram * 1 cm²

Now we need to convert the cubic centimeters to cubic meters to match the unit of acceleration (m/s²):

1 cubic centimeter = 0.000001 cubic meters

Work = 262,500,000 grams * 9.8 m/s² * 1 joule/gram * 0.000001 cubic meters

Simplifying the units:

Work = 2.565 * 10^6 joules

Therefore, the work required to pump the water out of the trough is approximately 2.565 million joules.

Now, let's address the second question - if the length of the trough is doubled, is the required work doubled?

The answer is no. Doubling the length of the trough does not double the work required. The work required to pump the water out is directly proportional to the volume of water in the trough, which is determined by the cross-sectional area (width and height) and the length. Doubling the length of the trough only increases the volume of water, but the relationship is not linear.