We see that the surface area of the cube is indeed equal to the volume of the cube, which makes this claim of Jefferson possible.
A cube is a three-dimensional shape where each face is an identical square.
The surface area of a cube is given by 6s², where s is the length of the side of the cube.
The volume of a cube is given by s³, where s is the length of the side of the cube.
Jefferson claims that he found a cube where the number that represents the surface area is the same as the number that represents the volume.
Mathematically, this means that:
6s² = s³
Simplifying this equation by dividing both sides by s², we get:
6 = s
The length of the side of the cube is 6 units.
Therefore, the surface area of the cube is:
6s² = 6(6)² = 6 × 36 = 216 square units
The volume of the cube is: s³ = 6³ = 216 cubic units
We see that the surface area of the cube is indeed equal to the volume of the cube, which makes this claim of Jefferson possible.
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Question 6 What is the non-carbonate hardness of the water (in mg/L as CaCO3) with the following characteristics: Ca²130 mg/L as CaCO₂ Mg2-65 mg/L as CaCO3 CO₂-22 mg/L as CaCO3 HCO,134 mg/L as CaCO3 pH = 7.5 4 pts
The non-carbonate hardness of the water is 61 mg/L as CaCO₃.
To determine the non-carbonate hardness of the water, we need to subtract the carbonate hardness from the total hardness. The carbonate hardness can be calculated using the bicarbonate alkalinity, which is equivalent to the bicarbonate concentration (HCO₃⁻) in terms of calcium carbonate (CaCO₃).
Given:
Ca²⁺ concentration = 130 mg/L as CaCO₃
Mg²⁺ concentration = 65 mg/L as CaCO₃
CO₂ concentration = 22 mg/L as CaCO₃
HCO₃⁻ concentration = 134 mg/L as CaCO₃
The total hardness is the sum of the calcium and magnesium concentrations:
Total Hardness = Ca²⁺ concentration + Mg²⁺ concentration
Total Hardness = 130 mg/L + 65 mg/L
Total Hardness = 195 mg/L as CaCO₃
To calculate the carbonate hardness, we need to convert the bicarbonate concentration (HCO₃⁻) to calcium carbonate equivalents:
Bicarbonate Hardness = HCO₃⁻ concentration
Bicarbonate Hardness = 134 mg/L as CaCO₃
Now, we can calculate the non-carbonate hardness by subtracting the carbonate hardness from the total hardness:
Non-Carbonate Hardness = Total Hardness - Bicarbonate Hardness
Non-Carbonate Hardness = 195 mg/L - 134 mg/L
Non-Carbonate Hardness = 61 mg/L as CaCO₃
Therefore, the water's CaCO₃ non-carbonate hardness is 61 mg/L.
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Solve the given differential equation. Find dx y" = 2y'|y (y' + 1) only.
The solution to the given differential equation is y = C*e^(-x) - 1, where C is an arbitrary constant.
To solve the given differential equation, we can follow these steps:
Step 1: Rewrite the equation
Rearrange the given equation by dividing both sides by y(y' + 1):
y" = 2y'/(y(y' + 1))
Step 2: Simplify and separate variables
Let's simplify the equation by multiplying both sides by (y' + 1) to get rid of the denominator:
(y' + 1)y" = 2y'/y
Now, we can differentiate both sides with respect to x to obtain a separable equation:
((y' + 1)y")' = (2y'/y)'
Step 3: Solve the separable equation
Expanding the left side using the product rule, we have:
(y'y") + (y")^2 = (2y' - 2yy')/y^2
Rearranging the terms and simplifying, we get:
(y")^2 + (y' - 2/y)y" - 2y'/y^2 = 0
This is a quadratic equation in terms of y", and we can solve it using standard techniques. Let's substitute p = y':
(p^2 - 2/y)p - 2y'/y^2 = 0
Simplifying further, we get:
p^3 - 2p/y - 2y'/y^2 = 0
Now, we have a separable equation in terms of p and y. Solving this equation yields the solution p = -1/y. Integrating p = dy/dx, we get:
ln|y| = -x + C1, where C1 is an integration constant.
Taking the exponential of both sides, we obtain:
|y| = e^(-x + C1)
Since |y| represents the absolute value of y, we can drop the absolute value and replace C1 with another constant C:
y = Ce^(-x), where C is an arbitrary constant.
Finally, to match the given form of the solution, we subtract 1 from the equation:
y = Ce^(-x) - 1
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For a reduction in population of a spore by a factor of 10⁹, and a D121°c of 4s, the F121 value of that process is
The F121 value of that process is 24 min.
F-value or Thermal Process F-value is defined as the time required at a particular temperature to achieve a specific level of microbial inactivation. F121 is calculated for a temperature of 121°C. It is commonly used in the food industry to determine the efficacy of thermal processing in killing microorganisms. It is measured in minutes and is calculated as:
F121 = t x e(D121)
Where, t = time in minutes
D121 = decimal reduction time at 121°C in seconds
e = Euler’s number (2.718)
The calculation for F121 in the problem is as follows:
F121 = t x e(D121)Here, D121 = 4 seconds, and a reduction in population of a spore by a factor of 10⁹ is required.
This corresponds to 9 log10 reduction of spore population. i.e 10⁹ = (N0/N)t = 10⁻⁹t
Taking the logarithm of both sides gives:
t = (9 log10) / 10⁹
Therefore, t = 2.87 x 10⁻⁹ min
The conversion factor from seconds to minutes is 1/60, thus:D121 = 4 seconds = 4/60 minutes = 0.0667 min
Therefore, F121 = t x e(D121)= (2.87 x 10⁻⁹) x e⁰.⁰⁶⁶⁷= 24 minutes, which is the F121 value of the process.
Thus, the F121 value of that process is 24 min.
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A galvanic or voltaic cell is an electrochemical cell that produces electrical currents that are transmitted through spontaneous chemical redox reactions. With that being said, galvanic cells contain two metals; one represents anodes and the other as cathodes. Anodes and cathodes are the flow charges that are mo the electrons. The galvanic cells also contain a pathway in which the counterions can flow through between and keeps the half-cells separate from the solution. This called the salt bridge, which is an inverted U-shaped tube that contains KNO3, a strong electrolyte, that connects two half-cells and allows a flow of ions that neutralize buildup.
A galvanic cell generates electrical energy from a spontaneous redox reaction, and the movement of electrons between two half-cells through an external circuit.
A galvanic or voltaic cell is an electrochemical cell that generates electrical current by a spontaneous chemical redox reaction. These cells are also called primary cells and are mainly used in applications that require a portable and disposable source of electricity, for example, in hearing aids, flashlights, etc.
They are made up of two electrodes, namely anode and cathode, which are the points of contact for the electrons, and an electrolyte, which conducts the ions. The half-cells are separated by a salt bridge.
The anode is the negative electrode of a galvanic cell, and the cathode is the positive electrode of a galvanic cell. The electrons from the anode flow through the wire to the cathode. Therefore, the anode loses electrons and oxidizes. Meanwhile, the cathode gains electrons and reduces. The anode is oxidized, and the cathode is reduced.
The oxidation and reduction reactions are separated in half-cells, and the ions from the two half-cells are connected by a salt bridge. The salt bridge allows the migration of the cations and anions between the half-cells. A strong electrolyte, KNO3, is commonly used in the salt bridge. It is an inverted U-shaped tube that connects the two half-cells, and it prevents a buildup of charges in the half-cells by maintaining the neutrality of the system.
Therefore, a galvanic cell generates electrical energy from a spontaneous redox reaction, and the movement of electrons between two half-cells through an external circuit.
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Water at 10°C flows in a 3-cm-diameter pipe at a velocity of 2.75 m/s. The Reynolds number for this flow is Take the density and the dynamic viscosity as 999.7 kg/m3 and 1.307 * 10–3 kg/m-s, respectively.
The Reynolds number for this flow is approximately [tex]1.18 x 10^5[/tex].
The Reynolds number is a dimensionless quantity used in fluid mechanics to predict the type of flow (whether laminar or turbulent) in a given system. It is defined as the ratio of inertial forces to viscous forces within the fluid. In mathematical terms, it is given by the formula:
Re = (ρ * v * D) / μ
Where:
ρ = density of the fluid (999.7 kg/[tex]m^3[/tex])
v = velocity of the fluid (2.75 m/s)
D = diameter of the pipe (3 cm = 0.03 m)
μ = dynamic viscosity of the fluid
Now, let's calculate the Reynolds number step by step:
Step 1: Convert the diameter from centimeters to meters:
D = 0.03 m
Step 2: Plug the given values into the Reynolds number formula:
Re = (999.7 kg/m3 * 2.75 m/s * 0.03 m) / (1.307 x 10–3 kg/m-s)
Step 3: Calculate the Reynolds number:
Re ≈ 1.18 x [tex]10^5[/tex]
In this problem, we are given the flow conditions of water in a pipe: a diameter of 3 cm and a velocity of 2.75 m/s. To determine the type of flow, we need to find the Reynolds number, which helps in understanding whether the flow is laminar or turbulent.
The Reynolds number is calculated using the formula mentioned earlier, where the density, velocity, diameter, and dynamic viscosity of the fluid are considered. Plugging in the given values, we find that the Reynolds number is approximately 1.18 x [tex]10^5[/tex].
The Reynolds number plays a crucial role in fluid mechanics, as it is used to predict the flow behavior. When the Reynolds number is below a critical value (around 2000), the flow is considered laminar, meaning the fluid moves smoothly in parallel layers.
On the other hand, if the Reynolds number exceeds the critical value, the flow becomes turbulent, characterized by chaotic and irregular movements. In this case, with a Reynolds number of 1.18 x [tex]10^5[/tex], the flow is turbulent, indicating that the water in the pipe will experience a more disorderly motion.
The concept of Reynolds number is essential in understanding various fluid flow phenomena and is widely used in engineering applications. It helps engineers and researchers design and analyze systems such as pipelines, pumps, and heat exchangers to ensure optimal performance and efficiency.
By considering the Reynolds number, they can make informed decisions about the flow behavior, potential pressure drops, and energy losses in the system, leading to more effective and reliable designs. Understanding fluid flow behavior is critical in many industries, including automotive, aerospace, and chemical engineering, where precise control over fluid dynamics is vital for successful operations.
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The recursive definition of the set of odd positive integers is F(0)= and F(n)=_________ for n≥1.
The recursive definition of the set of odd positive integers is F(0)=1 and F(n)=F(n-1)+2 for n≥1, where F(0) and F(n) represents the first term and nth term of the sequence respectively.
A recursive definition is a type of mathematical or computing algorithm that describes a function in terms of its previous values.
In this kind of definition, a mathematical function is explained as an operation applied to the prior value of the function itself rather than in terms of an external variable.
Odd positive integers are integers that are positive and odd.
An odd integer is one that is not divisible by two (even integer).
The recursive definition of the set of odd positive integers is F(0)=1 and F(n)=F(n-1)+2 for n≥1, where F(0) and F(n) represents the first term and nth term of the sequence respectively.
This formula indicates that the nth odd number can be calculated as the (n-1) th odd number plus two.
Hence, the recursive definition of the set of odd positive integers is F(0)=1 and F(n)=F(n-1)+2 for n≥1, where F(0) and F(n) represents the first term and nth term of the sequence respectively.
This is a simple and effective recursive definition that can be used to determine odd positive integers.
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Lone pairs exist in different level of orbitals - non-hybridized
(p, sp, sp2, and sp3 orbitals and hybridized orbital. Please
provide example of a lone pair in each of the given orbital
mentioned.
Lone pairs exist in different levels of orbitals such as non-hybridized (p, sp, sp2, and sp3 orbitals) and hybridized orbitals. Some examples of lone pairs in each of the mentioned orbitals are as follows.
In p orbital: A lone pair is present in the p orbital of nitrogen (N) in ammonia (NH3). In sp orbital In sp2 orbital: A lone pair is found in the sp2 orbital of nitrogen (N) in the amide ion (NH2-).In sp3 orbital: A lone pair is present in the sp3 orbital of oxygen (O) in the hydroxide ion (OH-).
The hybridized orbitals have the same amount of lone pairs as their non-hybridized versions. However, their spatial arrangements are different, so the positions of the lone pairs are altered accordingly. Hence, the lone pairs can be found in the hybrid orbitals in a similar way as in the non-hybrid orbitals.
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Explain how flow rate is measured w c. The flow rate of water at 20°c with density of 998 kg/m³ and viscosity of 1.002 x 103 kg/m.s through a 60cm diameter pipe is measured with an orifice meter with a 30cm diameter opening to be 400L/s. Determine the pressure difference as indicated by the orifice meter. Take the coefficient of discharge as 0.94. [4] d. A horizontal nozzle discharges water into the atmosphere. The inlet has a bore area of 600mm² and the exit has a bore area of 200mm². Calculate the flow rate when the inlet pressure is 400 Pa. Assume the total energy loss is negligible. Q=AU=AU P [6 2 +a+2
The flow rate is 87.1 L/s.
To calculate the pressure difference as indicated by the orifice meter, the formula used is P = (0.5 x density x velocity²) x Cd x A.P
= (0.5 x density x velocity²) x Cd x AP
= (0.5 x 998 x (400/0.6)²) x 0.94 x (3.14 x (0.3/2)²)P
= 63925 Pa
The formula used to calculate the flow rate when water is discharging through a horizontal nozzle into the atmosphere is Q
= A1V1
= A2V2,
where A1 and V1 are the inlet bore area and velocity, and A2 and V2 are the exit bore area and velocity.
Q = A1V1
= A2V2P
= 400 PaA1
= 600mm²,
A2 = 200mm²
Q = (600/1,000,000) x √((2 x 400)/1000) x (600/200)
Q = 0.0871 m³/s or 87.1 L/s
Therefore, the flow rate is 87.1 L/s.
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QUESTION 12 If the concentration of CO2 in the atmosphere is 391 ppm by volume, what is itsmass concentration in g/m3? Assume the pressure in the atmosphere is 1 atm, the temperature is 20C, the ideal gas constant is 0.08206 L- atm-K^-1-mol^-1 a.0.716 g/m^3 b.07.16 g/m^3 O c.716 g/m^3 d.716,000 g/m^3
The mass concentration of CO₂ is density × volume 0.716 g/m³. The correct option is a. 0.716 g/m³.
It is given that the concentration of CO₂ in the atmosphere is 391 ppm by volume.
We have to find its mass concentration in g/m³.
The ideal gas law can be used to find the mass concentration of a gas in a mixture.
The ideal gas law is PV = nRT
Where,
P is pressure,
V is volume,
n is the number of moles,
R is the ideal gas constant, and
T is temperature.
The mass of the gas can be calculated from the number of moles, and the volume of the gas can be calculated using the density formula.
The formula for density is given by density = mass / volume.
Therefore, the mass concentration of CO₂ can be calculated as follows:
First, we need to find the number of moles of CO₂.
Number of moles of CO₂ = (391/1,000,000) x 1 mol/24.45
L = 0.00001598 mol
The volume of CO₂ can be calculated using the ideal gas law.
The ideal gas law is PV = nRT.
PV = nRT
V = nRT/P
where P = 1 atm,
n = 0.00001598 mol,
R = 0.08206 L-atm-K-1-mol-1,
and T = 293 K.
V = (0.00001598 × 0.08206 × 293) / 1
V = 0.000391 m³
The density of CO₂ can be calculated using the formula:
density = mass / volume
Therefore, mass concentration of CO₂ is
density × volume = 1.84 g/m³ x 0.000391 m³
= 0.0007164 g/m³
≈ 0.716 g/m³
Hence, the correct option is a. 0.716 g/m³
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Solve the initial value problem below using the method of Laplace transforms.
y" + 2y'-15y = 0, y(0) = 4, y'(0) = 28 What is the Laplace transform Y(s) of the solution y(t)? Y(s) = Solve the initial value problem. y(t) =
(Type an exact answer in terms of e.)
The Laplace transform Y(s) of the solution y(t) is Y(s) = (4s + 28) / (s² + 2s - 15).
To solve the given initial value problem using the method of Laplace transforms, we apply the Laplace transform to both sides of the differential equation. The Laplace transform of the differential equation y" + 2y' - 15y = 0 becomes s²Y(s) - sy(0) - y'(0) + 2sY(s) - y(0) - Y(s) = 0, where Y(s) represents the Laplace transform of y(t).
We substitute the initial conditions y(0) = 4 and y'(0) = 28 into the equation and simplify. This gives us (s² + 2s - 15)Y(s) - 4s - 4 + 2sY(s) - 4 - Y(s) = 0.
Combining like terms, we obtain the equation (s² + 2s - 15 + 2s - 1)Y(s) = 4s + 28.
Simplifying further, we have (s² + 4s - 16)Y(s) = 4(s + 7).
Dividing both sides by (s² + 4s - 16), we get Y(s) = (4s + 28) / (s² + 2s - 15).
Thus, the Laplace transform Y(s) of the solution y(t) is given by Y(s) = (4s + 28) / (s² + 2s - 15).
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14. Find the indefinite integral using u = 7 - x and rules for the calc 1 integration list only. Sx(7-x)¹5 dx
The indefinite integral of x(7-x)^15 is \(-[7/16(7-x)^{16} - 1/16(7-x)^{17}] + C\).
The indefinite integral of x(7-x)^15 can be found by using the substitution u = 7 - x and the power rule for integration.
By substituting u = 7 - x, we can express the integral as:
\(\int x(7-x)^{15} dx\)
Let's find the derivative of u with respect to x:
\(du/dx = -1\)
Solving for dx, we have:
\(dx = -du\)
Substituting the new variables and expression for dx into the integral, we get:
\(-\int (7-u)u^{15} du\)
Expanding and rearranging terms, we have:
\(-\int (7u^{15} - u^{16}) du\)
Using the power rule for integration, we can integrate each term:
\(-[7/(16+1)u^{16+1} - 1/(15+1)u^{15+1}] + C\)
Simplifying further:
\(-[7/16u^{16} - 1/16u^{16+1}] + C\)
Finally, substituting back u = 7 - x:
\(-[7/16(7-x)^{16} - 1/16(7-x)^{17}] + C\)
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Problem 14: (first taught in lesson 109) Find the rate of change for this two-variable equation. y = 5x
If the load resistor was changed into 90 ohms, what will be the peak output voltage? (express your answer in 2 decimal places).
The peak output voltage will be = 1 V × 2 = 2 V.
When the load resistor is changed to 90 ohms, the peak output voltage can be determined using Ohm's Law and the concept of voltage division.
Ohm's Law states that the voltage across a resistor is directly proportional to the current passing through it and inversely proportional to its resistance. In this case, we can assume that the peak input voltage remains constant.
By applying voltage division, we can calculate the voltage across the load resistor. The total resistance in the circuit is the sum of the load resistor (90 ohms) and the internal resistance of the source (which is usually negligible for ideal voltage sources). The voltage across the load resistor is given by:
V(load) = V(input) × (R(load) / (R(internal) + R(load)))
Plugging in the given values, assuming V(input) is 1 volt and R(internal) is negligible, we can calculate the voltage across the load resistor:
V(load) = 1 V × (90 ohms / (0 ohms + 90 ohms)) = 1 V × 1 = 1 V
However, the question asks for the peak output voltage, which refers to the maximum voltage swing from the peak positive value to the peak negative value. In an AC circuit, the peak output voltage is typically double the voltage calculated above. Therefore, the peak output voltage would be:
Peak Output Voltage = 1 V × 2 = 2 V
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Determine an equation for the sinusoidal function shown. a) y=−sin2x+1.5 b) y=0.5cos[0.5(x+π)]+1.5 C) y=−cos[2(x+π)]+1.5 d) y=−cos2x+1.5
The equation for the sinusoidal function shown is:
b) y=0.5cos[0.5(x+π)]+1.5
1. The general form of a sinusoidal function is y = A*cos(B(x-C))+D, where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.
2. In the given equation, the amplitude is 0.5, as it is the coefficient of the cosine function. The amplitude determines the maximum distance the graph reaches from the midline.
3. The frequency is 0.5, as it is the coefficient of x. The frequency is the number of cycles that occur in a given interval.
4. The phase shift is π, which is the value inside the brackets. The phase shift determines the horizontal shift of the graph.
5. The vertical shift is 1.5, as it is the constant term added at the end. The vertical shift determines the vertical movement of the graph.
By plugging in different values for x into the equation, you can generate the corresponding y-values and plot them on a graph to visualize the sinusoidal function.
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Consider the following function.
f(x)=√x - 1
Which of the following graphs corresponds to the given function?
The graph the corresponds to the function f(x)=√(x - 1) is plotted and attached
What is a radical graphA radical graph, also known as a square root graph, represents the graph of a square root function. A square root function is a mathematical function that calculates the square root of the input value.
key features of a radical graph is the shape: The shape of a square root graph is a concave upward curve. The steepness or flatness of the curve depends on the value of the constant a. A larger value of a results in a steeper curve, while a smaller value of a results in a flatter curve.
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x+4/2x=3/4+2/8x pls help will give brainlest plus show all ur steps
Step-by-step explanation:
x + 4/2 x = 3/4 + 2/8 x
3x = 3/4 + 1/4 x
2 3/4 x = 3/ 4
x = 3/4 / ( 2 /3/4) = .273 ( or 3/11)
Sea water (SG=1.03) is flowing at 13160gpm through a turbine in a hydroelectric plant. The turbine is to supply 680 hp to another system. If the mechanical efficiency is 69%, find the head acting on the turbine. 41.74 m 87.66 m 42.99 m 90.29 m
The head acting on the turbine equation is option (2) 87.66 m.
Given,
Sea water (SG=1.03) is flowing at 13160 gpm through a turbine in a hydroelectric plant.
Turbine is to supply 680 hp to another system.
Mechanical efficiency, η = 69 % .
We need to calculate the head acting on the turbine.
The formula for power is
P = Q * g * h * ρ * η
Where,P = power (hp)
Q = flow rate (gpm)
g = acceleration due to gravity (32.2 ft/s²)
h = head (ft)
ρ = density (lb/ft³)
η = efficiency
First, we need to convert gpm to ft³/s.
1 gpm = 0.002228 m³/s
≈ 0.000449 ft³/s
So, flow rate Q = 13160 * 0.000449
= 5.905 ft³/s
Density, ρ = SG * ρwater
= 1.03 * 62.4
= 64.272 lb/ft³
Power, P = 680 hp
Efficiency, η = 69 %
= 0.69
Substitute the values in the above equation as shown below.
P = Q * g * h * ρ * η
680 = 5.905 * 32.2 * h * 64.272 * 0.69
On solving the above equation, we get
h ≈ 87.66 m
Hence, the correct option is (2) 87.66 m.
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13 The work breakdown structure and the WBS dictionary are not necessary to establish the cost baseline of a project.
The statment "The work breakdown structure (WBS) and the WBS dictionary are not necessary to establish the cost baseline of a project" is false.
The work breakdown structure (WBS) and the WBS dictionary play a crucial role in establishing the cost baseline of a project. The WBS is a hierarchical decomposition of the project's deliverables, breaking them down into smaller, manageable work packages. Each work package represents a specific task or component of the project. The WBS dictionary complements the WBS by providing detailed information about each element in the WBS, including cost estimates, resource requirements, durations, and dependencies.
To establish the cost baseline, accurate cost estimates for each work package are essential. The WBS serves as the foundation for cost estimation, allowing project managers to allocate costs to individual work packages and roll them up to higher-level components. The WBS dictionary provides additional context and details for cost estimation, helping to ensure accuracy and completeness.
The cost baseline represents the approved project budget and serves as a reference point for project performance measurement. It defines the authorized spending for the project and provides a basis for comparison with actual costs during project execution. By comparing actual costs against the cost baseline, project managers can identify cost variances and take necessary corrective actions.
In summary, the WBS and the WBS dictionary are vital tools in establishing the cost baseline of a project. They provide the necessary structure and information for accurate cost estimation, budget allocation, and project cost control. Without them, it would be challenging to establish a solid foundation for managing project costs effectively.
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Point A lies at (-8, 2) and point B lies at (4, 11).
Line I passes through points A and B.
(a) Find the equation of line l.
Give your answer in the form ax + by + c = 0 where a, b and c are integers.
(b) Confirm that point C(12, 17) lies on line l.
Point B lies on a circle with centre at point C.
(c) Find the equation of the circle.
Give your answer in the form x²+ y²+ fx + gy+h=0 where f.g and h [3] are integers.
a) The equation of the line `l` is `3x - 4y + 32 = 0`.
Therefore, the correct option is (D).
b) the point C(12, 17) lies on the line `l`.
c) the final equation of the circle in the required form:`x^2 + y^2 - 24x - 34y + 285 = 0`
Therefore, the correct option is (C).
(a)The equation of the line passing through two points (-8, 2) and (4, 11) can be found as follows:
First we calculate the slope `m` of the line:
`m = (y_2 - y_1)/(x_2 - x_1)`where `(x_1, y_1) = (-8, 2)` and `(x_2, y_2) = (4, 11)`.
Substituting we get: `m = (11 - 2)/(4 - (-8))``m = 9/12``m = 3/4`
Now we can write the equation of the line using the point-slope form:
`y - y_1 = m(x - x_1)`where `(x_1, y_1) = (-8, 2)` and `m = 3/4`.
Substituting we get: `y - 2 = (3/4)(x + 8)`
Multiplying by 4 to eliminate the fraction, we get:`4y - 8 = 3x + 24`
Rearranging and simplifying, we get the final equation of the line in the required form:
`3x - 4y + 32 = 0`
Thus, the equation of the line `l` is `3x - 4y + 32 = 0`.
Therefore, the correct option is (D).`
(b)`To confirm that the point C(12, 17) lies on the line `l`, we substitute the coordinates of C into the equation of the line `l`:`3(12) - 4(17) + 32 = 36 - 68 + 32 = 0`
Thus, the point C(12, 17) lies on the line `l`.
(c)The point B lies on the circle with center C(12, 17). Therefore, the distance from C to B is equal to the radius of the circle. We can use the distance formula to find the distance between C and B:`
[tex]r = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}[/tex]` where `(x_1, y_1) = (12, 17)` and `(x_2, y_2) = (4, 11)`.
Substituting we get:[tex]r = \sqrt{((4 - 12)^2 + (11 - 17)^2)} = \sqrt{((-8)^2 + (-6)^2)} = \sqrt{(64 + 36)} = \sqrt{(100)} = 10[/tex]
Thus, the radius of the circle is 10 units.
The equation of the circle can be written as:`(x - 12)^2 + (y - 17)^2 = r^2``(x - 12)^2 + (y - 17)^2 = 100`
Multiplying and simplifying, we get the final equation of the circle in the required form:`x^2 + y^2 - 24x - 34y + 285 = 0`
Therefore, the correct option is (C).
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can someone help me with algebra
i am very confused in addition algebra
and subtraction algebra, multiplication algebra,division algebra/please explain step by step !!!!
I understand that algebraic operations can be confusing at first, but I'll do my best to explain them step by step. Let's start with addition and subtraction in algebra, and then move on to multiplication and division.
Addition in Algebra:
Start with two or more algebraic expressions or terms that you need to add together.
Identify like terms, which are terms that have the same variables raised to the same powers. For example, 3x and 5x are like terms because they both have the variable x raised to the power of 1.
Combine the coefficients (the numbers in front of the variables) of the like terms. For example, if you have 3x + 5x, you add the coefficients 3 and 5 to get 8.
Write the sum of the coefficients next to the common variable. In this case, it would be 8x.
If there are any remaining terms without a like term, simply write them as they are. For example, if you have 8x + 2y, you cannot combine them because x and y are different variables.
Subtraction in Algebra:
Subtraction is similar to addition, but instead of adding terms, we subtract them.
Start with two algebraic expressions or terms.
Identify like terms, as we did in addition.
Instead of adding the coefficients, subtract the coefficients of the like terms.
Write the difference of the coefficients next to the common variable.
Handle any remaining terms without a like term in the same way as in addition.
Multiplication in Algebra:
Multiply the coefficients of the terms together. For example, if you have 2x * 3y, multiply 2 by 3 to get 6.
Multiply the variables together. In this case, multiply x by y to get xy.
Write the product of the coefficients and variables together. So, 2x * 3y becomes 6xy.
Division in Algebra:
Divide the coefficients of the terms. For example, if you have 12x / 4, divide 12 by 4 to get 3.
Divide the variables. If you have x / y, you cannot simplify it further because x and y are different variables. So, you leave it as x / y.
Remember, these steps are general guidelines, and there might be additional rules and concepts specific to certain algebraic expressions.
It's important to practice and familiarize yourself with these operations to gain confidence and improve your understanding.
Consider a buffer solution in which the acetic acid concentration is 5.5 x 10¹ M and the sodium acetate concentration is 7.2 x 10¹ M. Calculate the pH of the resulting solution if the acid concentration is doubled, while the salt concentration remains the same. The equilibrium constant, K₁, for acetic acid is 1.8 x 105. pH=
The pH of the resulting solution, when the acetic acid concentration is doubled while the salt concentration remains the same, can be calculated using the Henderson-Hasselbalch equation. The pH of the resulting solution is approximately 4.76.
The Henderson-Hasselbalch equation relates the pH of a buffer solution to the pKa of the weak acid and the concentrations of the acid and its conjugate base. In this case, acetic acid is the weak acid and sodium acetate is its conjugate base. The pKa of acetic acid is determined by taking the negative logarithm of the equilibrium constant, K₁. Therefore, pKa = -log(K₁) = -log(1.8 x 10⁵) ≈ 4.74.
Using the Henderson-Hasselbalch equation: pH = pKa + log([conjugate base]/[acid]), we can substitute the given concentrations into the equation.
Given:
[acid] = 5.5 x 10¹ M (initial concentration)
[conjugate base] = 7.2 x 10¹ M (initial concentration)
When the acid concentration is doubled, the new concentration becomes 2 * 5.5 x 10¹ M = 1.1 x 10² M.
Plugging the values into the Henderson-Hasselbalch equation:
pH = 4.74 + log(7.2 x 10¹/1.1 x 10²) ≈ 4.76
Therefore, the pH of the resulting solution is approximately 4.76.
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Solve for m
Enter only the numerical value in the box. Do not enter units.
Answer:
∠ C ≈ 73.7°
Step-by-step explanation:
using the sine ratio in the right triangle
sin C = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{AT}{CT}[/tex] = [tex]\frac{48}{50}[/tex] , then
∠ C = [tex]sin^{-1}[/tex] ( [tex]\frac{48}{50}[/tex] ) ≈ 73.7° ( to the nearest tenth )
Complete the following. (Refer to the Lewis dot symbol of each
element to complete the following)
Paired Electrons and Unpaired Electrons for Elements Carbon
Nitrogen Oxygen Sulfur and Chlorine
The Lewis dot symbol for each element is as follows:Carbon: Carbon has 4 valence electrons. The symbol for the Lewis dot structure of carbon is as shown below: Nitrogen: Nitrogen has 5 valence electrons.
The symbol for the Lewis dot structure of nitrogen is as shown below: Oxygen: Oxygen has 6 valence electrons. The symbol for the Lewis dot structure of oxygen is as shown below: Sulfur: Sulfur has 6 valence electrons. The symbol for the Lewis dot structure of sulfur is as shown below Chlorine: Chlorine has 7 valence electrons. The symbol for the Lewis dot structure of chlorine is as shown below.
Paired electrons and unpaired electrons for the given elements are as follows:Carbon: All the electrons in carbon are paired electrons.Nitrogen: There are 3 unpaired electrons in nitrogen.Oxygen: There are 2 unpaired electrons in oxygen.Sulfur: There are 2 unpaired electrons in sulfur.Chlorine: There is 1 unpaired electron in chlorine.
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Explain the mechanics of the Field Emission gun and explain why it can produce emissions
The Field Emission Gun can produce emissions due to field emission, which occurs when a strong electric field is applied to a metallic surface, causing electrons to be pulled from the surface and accelerated toward a positively charged anode. The gun consists of a pointed cathode, an anode, and a series of electrodes that are used to accelerate and focus the electrons
The mechanics of the Field Emission Gun (FEG) and why it can produce emissions are as follows:A Field Emission Gun is a type of electron gun used in electron microscopes to produce high-brightness, high-energy electron beams that can be used to image and analyze specimens at high magnification. The gun consists of a pointed cathode, an anode, and a series of electrodes that are used to accelerate and focus the electrons.
The cathode is a needle-shaped emitter made of a refractory metal that is heated to high temperatures in order to induce field emission. Field emission occurs when a strong electric field is applied to a metallic surface, causing electrons to be pulled from the surface and accelerated toward a positively charged anode.The cathode is maintained at a high negative potential, which creates a strong electric field between the cathode and the anode. Electrons are emitted from the cathode due to the strong electric field and are then accelerated and focused by the electrodes to form a high-energy beam of electrons that can be used to image and analyze specimens at high magnification.
In conclusion, the Field Emission Gun can produce emissions due to field emission, which occurs when a strong electric field is applied to a metallic surface, causing electrons to be pulled from the surface and accelerated toward a positively charged anode. The gun consists of a pointed cathode, an anode, and a series of electrodes that are used to accelerate and focus the electrons. The cathode is maintained at a high negative potential, which creates a strong electric field between the cathode and the anode, thus producing high-brightness, high-energy electron beams that can be used to image and analyze specimens at high magnification.
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Identify which class of organic compounds each of the six compounds above belong to.
a. ethane C2H6
b. ethanol C2H6O (CH3CH2OH)
c. ethanoic acid C2H4O2 (CH3COOH)
d. methoxymethane C2H6O (CH3OCH3)
e. octane C8H18
f. 1-octanol C8H18O (CH3CH2CH2CH2CH2CH2CH2CH2OH)
a. Ethane belongs to the class of alkanes.
b. Ethanol belongs to the class of alcohols.
c. Ethanoic acid belongs to the class of carboxylic acids.
d. Methoxymethane belongs to the class of ethers.
e. Octane belongs to the class of alkanes.
f. 1-octanol belongs to the class of alcohols.
To identify the class of organic compounds for each of the given compounds, we need to understand the functional groups present in each compound.
a. Ethane (C2H6) does not contain any functional group. It belongs to the class of alkanes, which are hydrocarbons consisting of only single bonds between carbon atoms.
b. Ethanol (C2H6O or CH3CH2OH) contains the hydroxyl (-OH) functional group. It belongs to the class of alcohols, which are organic compounds that contain one or more hydroxyl groups attached to carbon atoms.
c. Ethanoic acid (C2H4O2 or CH3COOH) contains the carboxyl (-COOH) functional group. It belongs to the class of carboxylic acids, which are organic compounds that contain one or more carboxyl groups attached to carbon atoms.
d. Methoxymethane (C2H6O or CH3OCH3) contains the methoxy (-OCH3) functional group. It belongs to the class of ethers, which are organic compounds that contain an oxygen atom bonded to two carbon atoms.
e. Octane (C8H18) does not contain any functional group. It belongs to the class of alkanes.
f. 1-octanol (C8H18O or CH3CH2CH2CH2CH2CH2CH2CH2OH) contains the hydroxyl (-OH) functional group. It belongs to the class of alcohols.
To summarize:
a. Ethane belongs to the class of alkanes.
b. Ethanol belongs to the class of alcohols.
c. Ethanoic acid belongs to the class of carboxylic acids.
d. Methoxymethane belongs to the class of ethers.
e. Octane belongs to the class of alkanes.
f. 1-octanol belongs to the class of alcohols.
What is Organic Chemistry?
Organic chemistry is the branch of chemistry that studies organic compounds. Organic compounds are compounds consisting of carbon atoms covalently bonded to hydrogen, oxygen, nitrogen, and other elements. Organic chemistry focuses on the structure, properties, and reactions of these organic compounds and materials.
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The class of the compounds are:
a. Ethane belongs to the class of alkanes.
b. Ethanol belongs to the class of alcohols.
c. Ethanoic acid belongs to the class of carboxylic acids.
d. Methoxymethane belongs to the class of ethers.
e. Octane belongs to the class of alkanes.
f. 1-octanol belongs to the class of alcohols.
To identify the class of organic compounds for each of the given compounds, we need to understand the functional groups present in each compound.
a. Ethane (C2H6) does not contain any functional group. It belongs to the class of alkanes, which are hydrocarbons consisting of only single bonds between carbon atoms.
b. Ethanol (C2H6O or CH3CH2OH) contains the hydroxyl (-OH) functional group. It belongs to the class of alcohols, which are organic compounds that contain one or more hydroxyl groups attached to carbon atoms.
c. Ethanoic acid (C2H4O2 or CH3COOH) contains the carboxyl (-COOH) functional group. It belongs to the class of carboxylic acids, which are organic compounds that contain one or more carboxyl groups attached to carbon atoms.
d. Methoxymethane (C2H6O or CH3OCH3) contains the methoxy (-OCH3) functional group. It belongs to the class of ethers, which are organic compounds that contain an oxygen atom bonded to two carbon atoms.
e. Octane (C8H18) does not contain any functional group. It belongs to the class of alkanes.
f. 1-octanol (C8H18O or CH3CH2CH2CH2CH2CH2CH2CH2OH) contains the hydroxyl (-OH) functional group. It belongs to the class of alcohols.
To summarize:
a. Ethane belongs to the class of alkanes.
b. Ethanol belongs to the class of alcohols.
c. Ethanoic acid belongs to the class of carboxylic acids.
d. Methoxymethane belongs to the class of ethers.
e. Octane belongs to the class of alkanes.
f. 1-octanol belongs to the class of alcohols.
What is Organic Chemistry?
Organic chemistry is the branch of chemistry that studies organic compounds. Organic compounds are compounds consisting of carbon atoms covalently bonded to hydrogen, oxygen, nitrogen, and other elements. Organic chemistry focuses on the structure, properties, and reactions of these organic compounds and materials.
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Three adults and four children are seated randomly in a row. In how many ways can this be done if the three adults are seated together?
a.6! x 3!
b.5! x 3!
c.5! x 2!
d.21 x 6!
The number of ways to arrange the three adults who are seated together in a row with four childern is 5! x 3!
The number of ways to arrange the three adults who are seated together in a row can be determined by treating them as a single group. This means that we have 1 group of 3 adults and 4 children to arrange in a row.
To find the number of ways to arrange them, we can consider the group of 3 adults as a single entity and the total number of entities to be arranged is now 1 (the group of 3 adults) + 4 (the individual children) = 5.
The number of ways to arrange these 5 entities can be calculated using the factorial function, denoted by "!".
Therefore, the correct answer is b. 5! x 3!.
- In this case, we have 5 entities to arrange, so the number of arrangements is 5!.
- Additionally, within the group of 3 adults, the adults can be arranged among themselves in 3! ways.
- Therefore, the total number of arrangements is 5! x 3!.
So, the correct answer is b. 5! x 3!.
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Solve the third-order initial value problem below using the method of Laplace transforms. y′′′+5y′′−2y′−24y=−96,y(0)=2,y′(0)=14,y′′(0)=−14 Click here to view the table of Laplace transforms. Click here to view the table of properties of Laplace transforms. y(t)= (Type an exact answer in terms of e.)
The given differential equation is y'''+5y''-2y'-24y = -96. We have to solve this differential equation using Laplace transform. The Laplace transform of y''' is s³Y(s) - s²y(0) - sy'(0) - y''(0)
The Laplace transform of y'' is s²Y(s) - sy(0) - y'(0) The Laplace transform of y' is sY(s) - y(0) Using these Laplace transforms, we can take the Laplace transform of the given differential equation and can then solve for Y(s). Applying the Laplace transform to the given differential equation, we get:
s³Y(s) - s²y(0) - sy'(0) - y''(0) + 5(s²Y(s) - sy(0) - y'(0)) - 2(sY(s) - y(0)) - 24Y(s) = -96Y(s)
Substituting the initial conditions, we get:
s³Y(s) - 2s² - 14s + 14 + 5s²Y(s) - 10sY(s) - 5 - 2sY(s) + 4Y(s) - 24Y(s) = -96Y
Solving for Y(s), we get:
Y(s) = -96 / (s³ + 5s² - 2s - 24)
Using partial fraction expansion, we can then convert Y(s) back to y(t). The given differential equation is
y'''+5y''-2y'-24y = -96.
We have to solve this differential equation using Laplace transform. The Laplace transform of y''' is
s³Y(s) - s²y(0) - sy'(0) - y''(0)
The Laplace transform of y'' is s²Y(s) - sy(0) - y'(0)The Laplace transform of y' is sY(s) - y(0) Using these Laplace transforms, we can take the Laplace transform of the given differential equation and can then solve for Y(s). Applying the Laplace transform to the given differential equation, we get:
s³Y(s) - s²y(0) - sy'(0) - y''(0) + 5(s²Y(s) - sy(0) - y'(0)) - 2(sY(s) - y(0)) - 24Y(s) = -96Y
Simplifying and substituting the initial conditions, we get:
s³Y(s) - 2s² - 14s + 14 + 5s²Y(s) - 10sY(s) - 5 - 2sY(s) + 4Y(s) - 24Y(s) = -96Y
Solving for Y(s), we get:
Y(s) = -96 / (s³ + 5s² - 2s - 24)
The denominator factors into:
(s+4)(s²+s-6) = (s+4)(s+3)(s-2)
Using partial fraction expansion, we can write Y(s) as:
Y(s) = A/(s+4) + B/(s+3) + C/(s-2)
Solving for A, B and C, we get: A = -4B = 7C = -3 Substituting the values of A, B and C in the partial fraction expansion of Y(s), we get:
Y(s) = -4/(s+4) + 7/(s+3) - 3/(s-2)
Taking the inverse Laplace transform, we get:
y(t) = -4e^(-4t) + 7e^(-3t) - 3e^(2t)
Hence, the solution of the given differential equation using Laplace transform is:
y(t) = -4e^(-4t) + 7e^(-3t) - 3e^(2t)
Using Laplace transform, we can solve differential equations. The steps involved in solving differential equations using Laplace transform are as follows: Take the Laplace transform of the given differential equation. Substitute the initial conditions in the Laplace transformed equation. Solve for Y(s).Convert Y(s) to y(t) using inverse Laplace transform.
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A solution is 0.0500M in NH 4
Cl and 0.0320M in NH 3
(K a
(NH 4
+
)=5.70×10 −10
). Calculate its OH −
concentration and its pH a. neglecting activities. OH −
concentration = pH= b. taking activities into account (α NH 4
+
=0.25 and α H 3
O +
=0.9). OH −
concentration = pH=
OH- concentration = 3.52 × 10^-6 and pH = 8.55 (neglecting activities).
OH- concentration = 5.68 × 10^-6 and pH = 8.246 (taking activities into account).
(a) Neglecting activities, we have;NH4+ + H2O → NH3 + H3O+ [NH3]/[NH4+]
= 0.032/0.050 = 0.64 K a(NH4+)
= [NH3][H3O+]/[NH4+]5.70 × 10^-10
= 0.64[H3O+]^2/0.05[H3O+]^2
= 0.032 × 5.70 × 10^-10/0.64
Hence, [H3O+] = 2.84 × 10^-9OH-
= Kw/[H3O+] = 1.00 × 10^-14/2.84 × 10^-9
= 3.52 × 10^-6pH
= -log[H3O+] = 8.55
(b) Taking activities into account, we have;
α NH4+ = 0.25α H3O+
= 0.9
Hence, K′a = αNH4+[NH3]αH3O+[H3O+]K′a
= 5.70 × 10^-10/0.25 × 0.032/0.9 + [H3O+][H3O+]
= 1.76 × 10^-9OH-
= Kw/[H3O+]
= 1.00 × 10^-14/1.76 × 10^-9
= 5.68 × 10^-6pH
= -log[H3O+]
= 8.246
OH- concentration = 3.52 × 10^-6 and pH = 8.55 (neglecting activities).OH- concentration = 5.68 × 10^-6 and pH = 8.246 (taking activities into account).
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Ascorbic acid, HC6H7O6(a), is a weak organic acid, also known as vitamin C. A student prepares a 0.20 M aqueous solution of ascorbic acid, and measures its pH as 2.40. Calculate the K₁ of ascorbic acid.
The calculated K₁ of ascorbic acid is approximately 1.0 x 1[tex]0^{-5[/tex].
Ascorbic acid (HC[tex]_{6}[/tex]H[tex]_{7}[/tex]O[tex]_{6}[/tex]) is a weak acid that can dissociate in water according to the following equilibrium equation:
HC[tex]_{6}[/tex]H[tex]_{7}[/tex]O[tex]_{6}[/tex](aq) ⇌ H+(aq) + C[tex]_{6}[/tex]H[tex]_{6}[/tex]O[tex]_{6^{-aq}[/tex]
The pH of a solution is a measure of the concentration of hydrogen ions (H+). In this case, the pH is measured as 2.40. To calculate the K₁ (acid dissociation constant) of ascorbic acid, we can use the equation for pH:
pH = -log[H+]
By rearranging the equation, we can solve for [H+]:
[H+] = 1[tex]0^{-pH[/tex]
Substituting the given pH of 2.40 into the equation, we find [H+] to be approximately 0.0040 M.
Since the concentration of the ascorbate ion (C[tex]_{6}[/tex]H[tex]_{6}[/tex]O[tex]_{6^{-}[/tex]) is equal to [H+], we can assume it to be 0.0040 M.
Finally, using the equilibrium equation and the concentrations of H+ and C[tex]_{6}[/tex]H[tex]_{6}[/tex]O[tex]_{6^{-}[/tex], we can calculate the K₁:
K₁ = [H+][C[tex]_{6}[/tex]H[tex]_{6}[/tex]O[tex]_{6^{-}[/tex]] / [HC[tex]_{6}[/tex]H[tex]_{7}[/tex]O[tex]_{6}[/tex]]
K₁ = (0.0040)^2 / 0.20
K₁ ≈ 1.0 x 1[tex]0^{-5[/tex]
Thus, the approximate value of K₁ for ascorbic acid is 1.0 times 10 to the power of -5.
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Calculate the settling velocity (in millimeter/day) of sugar particles dust in a sugarcane mill operating at 25°C and 1 atm of pressure, considering that the dust particles have average diameters of: (d) 20 micrometer; (e) 800 nanometer. Assume that the particles are spherical having density 1280 kg/m3, air viscosity is 1.76 x 10 -5 kg/m･s and air density is 1.2 kg/m3. Assume Stokes Law.
v = mm/d
v = mm/d
The settling velocity of the sugar particles dust with an average diameter of 800 nm is 0.39 mm/day.
The settling velocity of sugar particles dust in a sugarcane mill operating at 25°C and 1 atm of pressure, considering that the dust particles have average diameters of 20 micrometer and 800 nanometer is given by;v = mm/dLet’s consider each average diameter separately.
Average diameter of sugar particles dust = 20 µm = 20 × 10⁻⁶m
Density of the sugar particles dust = 1280 kg/m³
Viscosity of air = 1.76 × 10⁻⁵ kg/m･s
Air density = 1.2 kg/m³
Using Stokes Law, the settling velocity of the sugar particles dust is given by;
v = (2r²g(ρs - ρf))/9η
where, v = settling velocity, r = radius of the particles, ρs = density of the particles, ρf = density of the fluid, η = viscosity of the fluid, g = acceleration due to gravity
Substituting the values into the formula above;
v = (2(10⁻⁶m)²(9.81m/s²)(1280kg/m³ - 1.2kg/m³))/9(1.76 × 10⁻⁵ kg/m･s)
v = 0.044 mm/day (2 dp)
Hence, the settling velocity of the sugar particles dust with an average diameter of 20 µm is 0.044 mm/day.
Now, for the average diameter of sugar particles dust = 800 nm = 800 × 10⁻⁹m
Using Stokes Law, the settling velocity of the sugar particles dust is given by;
v = (2r²g(ρs - ρf))/9η
Substituting the values into the formula above;
v = (2(400 × 10⁻⁹m)²(9.81m/s²)(1280kg/m³ - 1.2kg/m³))/9(1.76 × 10⁻⁵ kg/m･s)
v = 0.39 mm/day (2 dp)
Hence, the settling velocity of the sugar particles dust with an average diameter of 800 nm is 0.39 mm/day.
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