12. (a) Is the subset { (e,0), (e,6), (e,12), (h,0), (h,6), (h,12) }, a subgroup of the direct product group ( V x Z18 )? (V is the Klein four group.) Carefully explain or justify your answer.

(b) Is the subgroup { (e,0), (e,6), (e,12), (h,0), (h,6), (h,12) }, a normal subgroup of the direct product group ( V x Z18 )? Carefully explain or justify your answer.

Explain how the performance of induction motor can be predicted by circle diagram. Draw the circle
diagram for a 3-phase, mesh-connected, 22.38 kW, 500-V, 4-pole, 50-Hz induction motor. The data
below give the measurements of line current, voltage and reading of two wattmeters connected to
measure the input :
No load 500 V 8.3 A 2.85 kW − 1.35 kW
Short circuit 100 V 32 A − 0.75 kW 2.35 kW

a. Using only P-v-T data, estimate the enthalpy of vaporization of water at T oC where, T is T= 38 and so on. b. If the enthalpy (h), as an unmeasured thermodynamic property is related to the temperature (T) and Pressure (P) as measured thermodynamic properties according to the following relation: h = f (T,P). Show that: dh = Cp dT +[ v – T (dv/dT)p] dP Where: Cp is the specific heat at constant pressure, v is the specific volume.

A positive charge q is fixed at the point x=0,y=0 and a negative charge -2q is fixed at the point x=a,y=0.

Part A:

Derive an expression for the potential V at points on the y-axis as a function of the coordinate y. Take V to be zero at an infinite distance from the charges.

Part B:

At which positions on the y-axis is V = 0?

Part C:

What does the answer to part A become when y>>a?

Part D:

Explain why this result is obtained.

Write a function that plots the following trajectory equations using python turtle graphics.

x = v*sin(A)*t

y = v*cos(A)*t -.5*(9.8)*t**2

Loop through t starting at 0 and increment by .1 until y<=0. (hint, while loop) Experiment with values of velocity V (1 to 20) and angle A (0 to 90) that gives you a good plot that does not go off the screen. Position the start of the plot at the left bottom of the screen

A spherical raindrop falling through mist builds up mass. The mass accumlates at a rate proportional to its cross-sectional area and invers;ey proportional to its velocity, that is, dm/dt=k?r2/v, where r is the radius of the raindrop at a given time and v is the downard speed at the same time.Assuming the density P of the rain drop is a constant. Ignoring the resistance do to the fog and the air, calculate the instaneous acceleration of the raindrop as a function of v,r,P,g and k.

A particle with charge 3.20×10¹⁹ C is placed on the x-axis in a region where the electric potential due to other charges increases in the +x direction but does not change in the y or z-direction.

Part A:
The particle, initially at rest, is acted upon only by the electric force and moves from point a to point b along the x-axis, increasing its kinetic energy by 1.60×10¹⁹J. In what direction and through what potential difference Vb-Va does the particle move?

a. The particle moves to the left through a potential difference of Vb-Va = 0.500 V.
b. The particle moves to the left through a potential difference of Vb-Va = -0.500 V
c. The particle moves to the right through a potential difference of Vb-Va = 0.500 V.
d. The particle moves to the right through a potential difference of Vb-Va = -0.500 V.
e. The particle moves to the left through a potential difference of Vb-Va = 5.00 V.
f. The particle moves to the right through a potential difference of Vb-Va = -5.00 V.

Part B:
If the particle moves from point b to point c in the y-direction, what is the change in its potential energy, Vc-Vb?

a. + 1.60×10¹⁹J
b. - 1.60×10¹⁹J
c. 0

Molarity of standard NaOH solution: x/1 H 2/3 Unknown + 2/3 NaOH -> 2/3 H2O +Na 2/3 Acetate

Vinegar:	Titration 1	Titration 2	Tirtration 3
Initial V of Acid	.01 mL	5.56 mL	11.32 mL
Final V of Acid	5.56 mL	10.71 mL	16.02 mL
Inital V of Base	.01 mL	27.12 mL	3.01 mL
Final V of Base	22.23 mL	44.71 mL	24.90 mL

Unknown Acid code and number of reactive hydrogens: H+3

Unknown Acid	Titration 1	Titration 2 (Error)	Titration 3	Titration 4
Mass of Acid	.1024 g	.1028g	.1021g	.1023g
Initial V of Base	24.90 mL	37.80 mL	45.5 mL	9.24 mL
Final V of Base	37.80 mL	45.5 mL	9.24 mL	17.80 mL

4. Find the molarity of the vinegar for each of your three vinegar titrations, then find the average molarity of the acid.

5. Find the percent deviations for your three vinegar molarities for the vinegar titrations.

6. From the moles of acetic acid for each vinegar titration, find the mass of acetic acid for that titration

Write a Python/NetworkX function add_weights(G1, G1), where G1 and G1 are intended to be graphs with exactly the same edges and such that each edge has either no attribute or a single attribute, ‘weight’, with a numerical value. It returns a graph, say, G3, that has the same edges as G1 and G2. Each edge e of G3 has a single attribute, ‘weight’, whose value is the sum of the ‘weight’ attributes of e in G1 and of e in G2 if ein G1 and e in G2 both have attribute ‘weight’. Otherwise, if e in one of G1 or G2 has attribute ‘weight’, then the value of ‘weight’ in G3 is the value of that ‘weight’ attribute. Otherwise (i.e., neither e in G1 nor e in G2 has attribute ‘weight’), the value of attribute ‘weight’ in e in G3is 0. This function returns G3.

Note that, where G is a graph with an edge (u, v), G[u][v] counts as False in a position expecting a Boolean value if the edge (u, v)in G has no edge attributes. This is useful in this problem since an edge either has a single attribute, ‘weight’, or has no attribute.

The following is test code, followed by its output.

if __name__ == "__main__"  :     

    Ga = nx.Graph()

    Ga.add_edges_from([(0, 1, {'weight': 2}), (1, 2, {'weight': 4}),

                      (2, 3), (3, 1, {'weight': 2}), (0, 3)])

    Gb = nx.Graph()

    Gb.add_edges_from([(0, 1, {'weight': 3}), (1, 2, {'weight': 5}),

                      (2, 3, {'weight':3}), (3, 1), (0, 3)])

           

    for u, v, attr in add_weights(Ga, Gb).edges(data=True):

        wGa = Ga[u][v]['weight'] if Ga[u][v] else None

        wGb = Gb[u][v]['weight'] if Gb[u][v] else None

        print("Edge ({}, {}): {} + {} = {}".format(u, v, wGa, wGb,

                                                    attr['weight']))

Output:

Edge (0, 1): 2 + 3 = 5

Edge (0, 3): None + None = 0

Edge (1, 2): 4 + 5 = 9

Edge (1, 3): 2 + None = 2

Edge (3, 2): None + 3 = 3

1). A group of students plans a tour. The charge per student is $66 if 25 students go on a trip. If more that 25 students participated, the charge per student is reduced by $2 times the number of students over 25. Find the number of students that will furnish the maximum revenue. The number of students that will furnish the maximum revenue is ______? The maximum revenue is $_______?

2) A motorboat is capable of traveling at a speed of 15 miles per hour in still water. On a particular​ day, it took 30 minutes longer to travel a distance of 18 miles upstream than it took to travel the same distance downstream. What was the rate of current in the stream on that​ day?