Piston rings are mass-produced. The target internal diameter is 45mm but records show
that the diameters are normally distributed with mean 45mm and standard deviation 0.05
mm. An acceptable diameter is one within the range 44.95 mm to 45.05 mm.
(i) What proportion of the output is unacceptable?
(ii) Above what diameter do the 40% largest diameters lie?

A piston?cylinder device initially contains 1.5?kg saturated liquid water at 120°C. Now heat is transferred to the water until the volume increases 100 times.

Determine:

a. Initial Volume of the water.

b. Final volume of water.

c. Final state of water.

d. Amount of heat added to the water

e. Draw the T?s diagram

0.5 kg of R12 are contained in a piston/cylinder and has a quality of 75% and a temperature of 0 C. Heat is transferred at constant pressure until the R12 is a saturated vapor. Determine: a) the initial volume of the R12; b) the work done; c) the heat transfer; d) the T-v diagram.

Ans:

a) 0.0208 m3 b) 2.13 kJ c) 18.9 kJ

A 10 cm -diameter cylinder contains argon gas at 10 atm pressure and a temperature of 60 ∘C . A piston can slide in and out of the cylinder. The cylinder's initial length is 19 cm . 2600 J of heat are transferred to the gas, causing the gas to expand at constant pressure. 1-What is the final temperature of the cylinder? 2-What is the final length of the cylinder?

A weighted piston-cylinder device (pressure is not constant) initially contains 1.2 kg saturated liquid water at 190oC. Now heat is transferred to the water until the volume quadruples (increase four times) and the cylinder contains saturated vapor only. Determine: a) The final volume of the tank. b) The final temperature and pressure. c) The internal energy change of the water.

The propane is contained in a piston-cylinder assembly, which maintains constant pressure at 0.2 MPa. If a mixture with a quality of 0.65 and a total volume of 0.07 m3 is slowly heated to a temperature of 54°C, a) what was the total work into the system b) what was the total amount of heat added to the system.

Please explain what tables you use to find the answer.

Air within a piston–cylinder assembly, initially at 15 lbf/ in.², 510°R, and a volume of 6 ft³, is compressed isentropically to a final volume of 1.75 ft³.


Assuming the ideal gas model with k = 1.4 for the air, determine the:

(a) mass, in lb.

(b) final pressure, in lbf/in.²

(c) final temperature, in °R.

(d) work, in Btu.

C program simple version of blackjack following this design.

1. The basic rules of game

A deck of poker cards are used. For simplicity, we have unlimited number of cards, so we can generate a random card without considering which cards have already dealt. The game here is to play as a player against the computer (the dealer). The aim of the game is to accumulate a higher total of points than the dealer’s, but without going over 21. The cards 2 to 10 have their face values as points. J, Q, and K are10 points each, and the Ace is either 1 point or 11 points (player's choice). To simplify the matter, we consider that the Ace is 11 points and we don’t have card J, Q, or K unless you like to implement the option anyway.

a)  Betting

The player first places a bet. Let’s assume the minimum bet is $10 and maximum = is $1000.

b) Each play will result in one of the following events for the player

• Lose -- the player's bet is taken by the dealer.
• Win -- the player wins as much as the bet. If s/he bet $= 10, s/he wins $10 from the dealer.
• Blackjack - the player wins 1.5 times the bet. With a bet of $10, s/he wins $15 from the dealer. To simplify the matter, you can ignore Blackjack.
• Push - the hand is a draw. The player keeps his/her bet, neither winning nor losing money.

c) The start of the game

At the start, the player and the dealer receive two cards each. The player’s cards are normally dealt face up (displayed), while the dealer has one face down (called the hole card) and one face up. The best possible blackjack hand is an opening deal of an Ace with any of the ten-point cards. This is called a "blackjack", or a natural 21, and the player holding this automatically wins unless the dealer also has a blackjack. If a player and the dealer each have a blackjack, the result is a push.

d)   The player’s turn

The player can keep his hand as it is (stand) or take more cards from the deck (hit), one at a time, until either the player judges that the hand is strong enough to go up against the dealer’s hand and stands, or until it goes over 21, in which case the player immediately loses (busted).

e) The dealer’s turn

The dealer turns over the hidden hole card. The dealer hits (takes more cards) or stands depending on the value of the hand. The dealer must hit if the value of the hand is lower than 17, otherwise the dealer stands.

If the dealer is busted, the player wins. Otherwise the player wins if s/he has a higher score, loses if s/he has a lower score, or pushes if s/he has the same score as the dealer.

Blackjack consideration is not required, unless you like to implement the option anyway. By the way, a blackjack hand beats any other hand, also those with a total value of 21 but with more cards (which is not a natural blackjack).

f)  The program towards the end

If the player won or lost, s/he must decide whether to quit or to play another game unless the player runs out of money. Your program should give the player an initial betting amount of $1000.00.

2. The specific design of this project

a)  The main() program and its variables

You will need to decide on appropriate variables in which to store the player's bankroll (in order to keep track of how much money or how many points the player has), the bet at a game, and other information. Let’s use an integer array gamerecord[] to store how many times the player won, lost, hit a blackjack, and got busted. (Again, blackjack is optional).

The bankroll, bet, and gamerecord[] should be kept up to date on the player's current status. (The program calls playing() to play a game, as discussed below. )

After each game, the program must report the result of the game: the amount of money won or lost, the current value of the bankroll, how many times the player won and lost, and how many times the player hit a blackjack and got busted. (You may want to record and report how many times the dealer got busted as well, as an option.)

After each game (by calling playing()), the program should allow the player to continue playing until s/he chooses to quit, or until s/he runs out of money. This central program control may be done within main(), in a do-while loop: 1) call playing() to play a game; 2) check whether to play again. We will add some more components later.

b) “Dealing” the card: the dealing() function

A separate dealing() function will be used to generate a card number. You may want to implement and double check this function first. You will use a random number generator. The random number generator needs to be seeded with the current time at the beginning of the main program. The possible random values generated are 1 to 10 (or 13 if J, Q, and K are considered), representing the cards’ face values. This function will return the number generated. The return value 1 represent the Ace’s face value (and the return value 11, 12, and 13 are J, Q, and K’s face value, respectively.) A large random number n can be converted to a value between 1 to 13 by:  (1 + n%13).

c) “Playing” the game: playing() function

A second function playing() will be used to play a single game until the player either wins or loses a bet, based upon the rules given above. This function should get a bet, modify the current amount of the player's bank roll according to the game result, modify the gamerecord array values of the player won or lost, and the player hit the blackjack or got busted.  These values are returned through function parameters by address passing in playing().

Within the function, the player is asked to place a bet (10 to 1000 within the bankroll amount), so the corresponding value is read from the keyboard. The system (dealer) then "deals the cards" (simulated by calling the function dealing(), one card at a time). After each dealing, this function should report the card values, except the dealer’s hole card. The function should have two variables to store the player and the dealer’s scores. Remember face value 1 represents score 11 (or 1 if you want to be more complete as an option, and 11, 12, or 13 represents score 10).

The player can keep his hand as it is (stand) or take more cards from the deck (hit), one at a time, until either the player judges that the hand is strong enough to go up against the dealer's hand and stands, or until it goes over 21, in which case the player immediately loses the bet.

The dealer turns over his hidden hole card by displaying the hold card face value, and starts the game process automatically until the dealer wins or loses.

d) "Ending" and "Beginning" of the game

This part is implemented after you have done your programming as described above already.

You need a separate function ending() to do the following: you should report the current value of the bank roll, how many times the player won, lost, hit a blackjack, and went busted. You need to save the above information into a text file as well.

You need a separate function beginning() to do the following at the beginning of your program in main(): the function will open the text file you used to save the game information for reading if it exists, so that your game can continue from previous played results. If the file does not exist or the bank roll has a balance below the minimum bet, you start the game from scratch as usual, and report “new game” or “continual game”.

So,  main() includes 1) beginning();  2) a loop: playing(); 3) ending();

When we toss a penny, experience shows that the probability (longterm proportion) of a head is close to 1-in-2. Suppose now that we toss the penny repeatedly until we get a head. What is the probability that the first head comes up in an odd number of tosses (one, three, five, and so on)? To find out, repeat this experiment 50 times, and keep a record of the number of tosses needed to get a head on each of your 50 trials.

(a)

From your experiment, estimate the probability of a head on the first toss. What value should we expect this probability to have?

b)

Use the expected value to estimate the probability that the first head appears on an odd-numbered toss.

1. Alice thinks that the more often you read or watch the news, the more likely you are to vote. She asks five people how many days per week they read or watch the news (X) and how likely they are to vote on a scale from 1 to 10 (Y).
a. (14 points) Calculate the correlation between the two variables:
X Y
3 2
2 1
1 3
5 8
6 9

b. (2 points) How much of the variability in likelihood to vote is explained by frequency of reading or watching the news?

2. Over the years, Coach Bob has developed a formula to predict how many of the country’s top 100 football recruits will sign with his university based on how many games his team won that year. The formula is:
Y = 2(X) – 14
a. (6 points) How many top recruits would be predicted for seasons that ended with:
12 wins?
10 wins?
7 wins?
b. (2 points) What is the predictor variable? What is the criterion variable?
c. (2 points) What is the slope? What is the y-intercept?
d. (2 points) Is the correlation between games won and number of top recruits signed positive or negative? How do we know?

3. (6 points) Calculate the standard error for the following samples. The population standard deviation for each sample is 15.
a. N = 4
b. N = 16
c. N = 225

4. a. (3 points) If we know the population mean is 50 and the standard error of the mean is .5, what is the z-score for a sample mean of 51? What is the likelihood of getting a sample mean of 51 or more?   

b. (3 points) If we know the population mean is 20 and the standard error of the mean is 5, what is the z-score for a sample mean of 105? What is the likelihood of getting a sample mean of 105 or higher？