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 out the work required to pump out the water from the trough, we need to calculate the volume of water in the trough first. The volume of a trapezoidal cross section can be calculated using the formula:
V = (a + b) / 2 * h * L
Where:
V = Volume of water
a = Length of the top side of the trapezoid (1 m)
b = Length of the bottom side of the trapezoid (1/2 m)
h = Height of the trapezoid (1 m)
L = Length of the trough (10 m)
Substituting the given values into the formula, we get:
V = (1 + 1/2) / 2 * 1 * 10
V = (3/2) / 2 * 10
V = (3/4) * 10
V = 7.5 m^3
Now, the work required to pump out the water can be calculated using the formula:
W = m * g * h
Where:
W = Work required
m = Mass of the water (density * volume)
g = Acceleration due to gravity (approximately 9.8 m/s^2)
h = Height from which the water is lifted (will assume 1 m)
The density of water is approximately 1000 kg/m^3. So, the mass of the water can be calculated as:
m = density * V
m = 1000 * 7.5
m = 7500 kg
Now, substituting the values into the formula for work, we get:
W = 7500 * 9.8 * 1
W = 73,500 J
Therefore, the work required to pump out the water in the trough is 73,500 J.
To answer the second question (B) - if the length of the trough is doubled, the amount of work needed does not double. The work required to pump out the water depends on the height from which the water is lifted (h), not the length of the trough. So, as long as the height remains the same, the work required will remain the same, regardless of the length of the trough.
To find the amount of work required to pump out the water in the trough, we first need to calculate the volume of water in the trough. The formula for the volume of a trapezoidal prism is:
V = ((a + b) / 2) * h * l
Where:
- V is the volume
- a and b are the horizontal sides of the trapezoid
- h is the height of the trapezoid
- l is the length of the trough
Given the following measurements:
- a = 1/2 m
- b = 1 m
- h = 1 m
- l = 10 m
We can substitute these values into the formula to calculate the volume:
V = ((1/2 + 1) / 2) * 1 * 10
V = (3/2) * 10
V = 15 m³
Now that we know the volume of water in the trough is 15 cubic meters, we can proceed to calculate the work required to pump it out. The formula for work is:
W = F * d
Where:
- W is the work
- F is the force required to lift the water
- d is the distance over which the force is applied
In this case, the force required to lift the water is equal to the weight of the water, given by:
F = m * g
Where:
- F is the force
- m is the mass of the water
- g is the acceleration due to gravity (approximately 9.8 m/s²)
The mass of the water can be calculated as:
m = V * ρ
Where:
- m is the mass
- V is the volume of water
- ρ is the density of water (approximately 1000 kg/m³)
Substituting the values, we have:
m = 15 * 1000
m = 15000 kg
Now we can calculate the force:
F = 15000 * 9.8
F = 147000 N
Finally, we need to determine the distance the force is applied over. In this case, it is the height of the trough, given as 1 m:
d = 1 m
Substituting the values, we can now calculate the work required:
W = 147000 * 1
W = 147000 J
So, the amount of work required to pump out the water in the trough is 147,000 Joules.
Now, let's move on to part B: If the length is doubled, does the amount of work needed double?
To answer this question, we need to examine the formulas we used. The work formula is:
W = F * d
We see that the work depends on both the force (F) and the distance (d). In this case, doubling the length of the trough would only affect the distance over which the force is applied. Since the force required (F) remains the same (assuming all other dimensions remain constant), doubling the length will indeed double the distance. Therefore, the amount of work required would also double.
In summary, if the length of the trough is doubled, the amount of work needed to pump out the water will also double.