A metal cylinder of mass ? = 5 kg with a coaxial hole is sliding horizontally on a metal rod. A spring with spring constant ? = 500 N/m is attached to the cylinder. The other end of the spring is attached on a table at a vertical distance of ℎ = 0.3 m below the cylinder. The spring has an equilibrium length of ?_0 = 0.5 m.

a) What is the horizontal position ?0 of the cylinder when the spring is at its natural equilibrium length of ?0 = 0.5 m? Define the attachment point of the spring on the table as position ? = 0.

b) What is the length ? of the spring when the cylinder is at any given position ??

c) Draw a free-body diagram of the cylinder.

d) Show that the horizontal force from the spring on the cylinder is: ?? = −?? (1 − ?0 √? 2 + ℎ 2 )

Microsoft has an equity value of $250B, but has only $10B in debt. Management decides they’d like to increase the D/(D+E) ratio to 40% and use the proceeds to repurchase shares of equity, i.e. a leveraged recapitalization.

a) Before the recapitalization, the debt is considered risk-free and the equity has a beta of 1.13; afterwards, the debt beta becomes 0.3. How much debt does Microsoft issue? By how much has this restructuring changed the riskiness of equity in Microsoft, i.e. what is
the equity beta after the recapitalization?

b. If the risk-free rate is 3% and the expected market return is 9%, what are the expected
returns to equity holders before and after the recapitalization?

Please answer all parts of this question.

Drew works at Baseball Inc., a seller of high-quality baseballs. He is interested in forecasting demand for his baseballs that are sold weekly to University of Miami using exponential smoothing.

Assume an initial forecast of 175 and the demand data below:

You want to use simple exponential smoothing. In case 1 you take alpha = 0.7 and in case 2 you take alpha = 0.3. Which of the following has an acceptable tracking signal, i.e., model does not need to be re-evaluated?

Answer Options:

1. Case 1
2. Case 2
3. Both Case 1 and Case 2
4. Neither Case 1 and Case 2

A 50 Hz, four-pole, 480 V, three-phase, wye-connected induction machine
has the following parameters:
R1 = 0.2 R2 = 0.3 Xeq = 12
The motor is loaded by a unidirectional constant-torque load of 40 NM. A
countercurrent braking is performed on the motor. Calculate the following:
a. Motor current before the countercurrent braking
b. Motor current just after the voltage sequence is reversed
C. Current at the moment when the motor stops
d. Moter torque at the condition of part ( b) and (c).
e. Motor current at the new steady state point
F. Motor torque at the new steady state point
g. Motor speed at the new steady-state point

Please help answer the question quickly

The Neal Company wants to estimate next year's return on equity (ROE) under different financial leverage ratios. Neal's total capital is $18 million, it currently uses only common equity, it has no future plans to use preferred stock in its capital structure, and its federal-plus-state tax rate is 40%. The CFO has estimated next year's EBIT for three possible states of the world: $4.6 million with a 0.2 probability, $2.4 million with a 0.5 probability, and $0.4 million with a 0.3 probability. Calculate Neal's expected ROE, standard deviation, and coefficient of variation for each of the following debt-to-capital ratios. Do not round intermediate calculations. Round your answers to two decimal places at the end of the calculations.

1.Plotting densitiesPlot the probability mass function (pmf) or probability density function (pdf) for eachof the following scenarios:(a) Consider abinomialrandom variable,X.i. Plot the pmf ofX∼Bin(n= 10,p= 0.3).ii. Plot the pmf ofX∼Bin(n= 10,p= 0.7).iii. Plot the pmf ofX∼Bin(n= 100,p= 0.3).iv. What happens to the shape of the pmf ofX∼Bin(n,p) whenpgets larger?v. What happens whenngets larger?(b) Consider ageometricrandom variable,Y.i. Plot the pmf ofY∼Geom(p= 0.1).ii. Plot the pmf ofY∼Geom(p= 0.5).iii. Plot the pmf ofY∼Geom(p= 0.8).iv. What happens to the shape of the pmf ofY∼Geom(p) whenpgets larger?(c) Consider aexponentialrandom variable,T.i. Plot the pdf ofT∼Exp(λ= 0.1).ii. Plot the pdf ofT∼Exp(λ= 0.5).iii. Plot the pdf ofT∼Exp(λ= 2).iv. What happens to the shape of the pdf ofT∼Exp(λ) whenλgets larger?(d) Consider anormalrandom variable,M.i. Plot the pdf ofM∼N(μ= 2,σ2= 1).ii. Plot the pdf ofM∼N(μ=−1,σ2= 1).iii. Plot the pdf ofM∼N(μ= 2,σ2= 5).1 iv. What happens to the pdf ofM∼N(μ,σ2) whenμis changed?v. What happens to the pdf ofM∼N(μ,σ2) whenσ2gets larger?(e) Which of the continuous distributions looks the most similar to the geometricdistribution? Which looks the most similar to the binomial distribution (withlargen)? Do these relationships make sense, based on your knowledge of thedistributions and their assumptions?

40)

In order to analyze water samples using a spectrophotometer or plate reader, it is necessary to turn the molecules of nitrate into a dye molecule that can be quantified. The first step in turning nitrate (NO₃^-)  into a dye molecule is reducing it to a molecule of nitrite (NO₂^-). This is done by reacting the NO₃^- with cadmium.

After the reduction reaction, the NO₂^- is reacted with two additional reagents. The first reagent, Reagent A, is a solution of sulfanilamide and hydrochloric acid. The second reagent, Reagent B, is a solution of N-(1-naphthyl)-ethylenediamine, called NNED for short. The compounds are mixed with the water sample and produce a purple color. The intensity of the purple color is directly related to the concentration of nitrite in the water sample. We can measure how purple the water turns as absorbance on a spectrophotometer and then convert the absorbance to concentration of nitrate.

To make Reagent A, we will need to make a solution of 10.0 g of sulfanilamide in 1 L of 2.4 molar hydrochloric acid (HCl).

The stock solution of HCl is 12 molar HCl. How many milliliters (mL) of 12 M HCl would you add to produce 0.3 liters (L) of 2.4M HCl?  mL HCl

After creating 0.3 L of 2.4 molar HCl solution, how many grams of sulfanilamide will be added?  

g sulfanilamide

41) After reacting the nitrate with cadmium to produce nitrite, the nitrite is then reacting with sulfanilamide and N-(1-naphthyl)-ethylenediamine, to produce a purple dye molecule that can be quantified on a spectrophotometer. The N-(1-naphthyl)-ethylenediamine, called NNED for convenience, reagent is made by mixing 1 gram of NNED in 1 liter of water. However, we don't always want to make an entire liter of solution because the NNED solution only lasts about 1 month before going bad and turning brown. How many milligrams of NNED will need to be added to make 0.15 liters of solution?

A young engineering student tries to decide what she will do on Sunday evening. She has three options; going to cinema (C), going to bowling saloon (B), going to a restaurant (R). She may choose C option with probability 0.4, B option with probability 0.2 and R option with probability 0.4. If she chooses cinema option, she may go to three movies; a thriller (CT), a cartoon (CC) and romantic comedy (CR). Conditional probabilities associated with these three movie options are 0.6, 0.35 and 0.05. If she chooses to go to bowling saloon, she has two options; Rocker House (BR) with probability 0.7 and Empire Bowling (BE) with probability 0.3. Restaurant options are Chinese (RC) with probability 0.1, Burger House (RB) with probability 0.2 and Urfa Kebap (RU) with probability 0.7. a) (16 Points) Draw the decision trees diagram for these options.A young industrial engineering student tries to decide what she will do on Sunday evening. She has three options; going to cinema (C), going to bowling saloon (B), going to a restaurant (R). She may choose C option with probability 0.4, B option with probability 0.2 and R option with probability 0.4. If she chooses cinema option, she may go to three movies; a thriller (CT), a cartoon (CC) and romantic comedy (CR). Conditional probabilities associated with these three movie options are 0.6, 0.35 and 0.05. If she chooses to go to bowling saloon, she has two options; Rocker House (BR) with probability 0.7 and Empire Bowling (BE) with probability 0.3. Restaurant options are Chinese (RC) with probability 0.1, Burger House (RB) with probability 0.2 and Kebap (RK) with probability 0.7

. a) Draw the decision trees diagram for these options.

b) Compute the probabilities for whole options