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.

To find the amount of work required to pump out the water from the trough, we need to calculate the volume of water in the trough and then multiply it by the weight of the water. The weight of the water can be found using its density and volume.

A. Let's start by finding the volume of water in the trough. The trough has a trapezoidal cross-section, so we can use the formula for the area of a trapezoid to find the cross-sectional area at the bottom of the trough:

Area_bottom = (1/2) × (0.5 + 1) × 1 = (0.5 + 1) × 1 = 1.5 m²

The volume of water in the trough can then be obtained by multiplying the cross-sectional area by the length of the trough:

Volume = Area_bottom × Length = 1.5 m² × 10 m = 15 m³

Now, let's find the weight of the water. The density of water is approximately 1000 kg/m³:

Weight = Density × Volume = 1000 kg/m³ × 15 m³ = 15000 kg

To pump the water out, we need to perform work against the gravitational force. The amount of work required is given by the equation:

Work = Force × Distance

Since the force required to lift an object is equal to its weight, and the distance here is the height of the trough (1 m), the work required can be calculated as:

Work = Weight × Height = 15000 kg × 1 m = 15000 kg·m

So, the work required to pump out the water from the trough is 15000 kg·m.

B. If the length of the trough is doubled, the amount of work needed does not double. The amount of work required is directly proportional to the weight of the water, which is determined by its volume and density. Since the volume of water is proportional to the length of the trough, doubling the length will double the volume of water. Therefore, the amount of work required will also double.