2. A pendulum exhibits simple harmonic motion when it is released at an angle θ = 9°. The length of the pendulum is l=0.86 m, and the pendulum bob has a mass m=0.3 kg.

a. What is the pendulum’s frequency?

b. Use energy conservation to find the pendulum bob’s speed when it passes through the lowest point of the swing? Assume that the difference in height between the lowest part of the swing and the highest part of the swing is delta H= l(1-cos(θ))

c. What is the total energy of the system?

a) Do the following production functions exhibit constant returns to scale, increasing returns to scale, or decreasing returns to scale? For full credit, show why.

1) Q= 10L^ 0.5K^0.3

2) Q= 10L^0.5K^0.5

3) Q= 10L^0.5K^0.7

4) Q= min{K, L}

b) Which objects pin down a_LC and a_KC? Explain carefully.

c) Why does labor being mobile across sectors automatically imply revenue maximization for firms? Explain carefully.

Assume that there was a starburst that produced 10¹¹ solar masses of stars with a minimum mass of 0.3 solar masses and a maximum mass of 100 solar masses. The number of stars as a function of mass is N(M) = KM^-2.35, where K is a normalization constant. The current turnoff mass of the population is 1 solar mass

1. Compute the number of stars and the total mass of stars that are still on the main sequence.
2. Assume that all stars with masses between 1 and 8 solar masses became white dwarfs. Compute the total number of white dwarfs in this population

Assume in a Bertrand model, there are 3 identical firms in the market with the same constant marginal cost of 30. The demand function is P = 150 ? Q. Firms share the monopoly profit equally if they participate in the cartel. With probability 0.3 the cartel is detected by the end of period t, in which case each cartel member has to pay the fine $1000. Cartel investigation takes one period. If detected by the end of period t, the cartel is disbanded at the end of the period and the firms return to the noncooperative game permanently. Derive the condition required for the cartel to succeed

Suppose that we have a single input variable X and all possible values of X is {0, 1, 2}. Also, suppose that we have two groups for outcomes (i.e., Y = 1, 2). It is known that the X|Y = 1 has Binomial(2, 0.7) and X|Y = 2 has the discrete uniform. In addition, P(Y = 1) = 0.3.

(1) Predict Y for X = 0, 1, 2, respectively, using the Bayes classifier.

(2) Compute the overall Bayes error rate.

Suppose we have a two-armed bandit. Our estimate of the payout rate of the first arm is 0.7, and our estimate of the payout rate for the second arm is 0.6. That is, ρ1 = 0.7 and ρ2 = 0.6. Our 95% confidence intervals for θ1 and θ2 are (0.6, 0.8) and (0.3, 0.9), respectively. Suppose you used an -greedy strategy with = 0.5. How might you decide which arm to pull? Suppose you used an interval exploration strategy with 95% confidence intervals. What arm would you pull?

0.2 kg/s of moist air at 45oC (DBT) and Humidity ratio 0.006 kgw/kgda (dry air) is mixed with 0.3 kg/s of moist air at 250C (DBT) and a humidity ratio of 0.018 kgw/kg da in an adiabatic mixing chamber. After mixing, the mixed air is heated to a final temperature of 40o C using a heater. Assume the barometric pressure to be 1 atm. Find

Enthalpies of both fluids

Temperature and humidity ratio of air after mixing.

Heat transfer rate in the heater.

Consider 0.8 kg of N₂ at 300 K, 1 bar contained in a rigid tank connected by a valve to another rigid tank holding 0.3 kg of CO₂ at 300 K, 1 bar. The valve is opened and gases are allowed to mix, achieving an equilibrium state at 290 K.
Determine:
(a) the volume of each tank, in m³.
(b) the final pressure, in bar.
(c) the magnitude of the heat transfer to or from the gases during the process, in kJ.
(d) the entropy change of each gas and of the overall system, in kJ/K.

The management of TTT Co must choose whether to go ahead with either of two mutually exclusive projects, A and B. The expected profits are as follows.

Required:

(a) Based on expected values, what would be the decision (project A or project B), if no information about demand were available.

(b) Calculate the value of perfect information about demand.