15) In the game blackjack, face cards are worth 10 points, aces are worth 1 or 1 1, and all other cards are worth their face value. You are dealt two cards with the objective of getting more points than the dealer. A "Blackjack" is 21. Assuming a fresh deck (i.e. no cards have been dealt), what are the odds of getting blackjack? 16) Assuming two decks of cards (again, assume a fresh deck), if the dealer is showing an ace, and you have not drawn any additional cards yet, what are the odds that the dealer has blackjack? . .. 17) Suppose that you are playing craps. If you roll the dice 10 times, what are the odds that 4 of your rolls come up with a total of seven? .L. .13) We need to enclose a field with a fence. We have 500 feet of fencing and a building is on one side and so won't need any fencing. Determine the dimensions of the field that will enclose the largest area. Building X Field y So, the problem we have is: So, set up the lagrangian take the derivatives with respect to x and y and set them equal to zero From the second equation, we have x = 2. Plug into the first equation to get 14) Suppose that Apple is selling IPads in both the US and Europe. Sales in each country are a function of the level of advertising and given by Sus =12+6Aus -1.2Ads SE =8+2AF -.2A Solve Apples' maximization problem; maximize total sales across the two districts subject to a total advertising budget of $4M. How would a $1M increase in Apples' advertising budget influence sales?11) Suppose that you are a pizza shop. Your profits depend on your sales of pizza and beer. Specifically, your profits as a function of Pizza sales (P) and beer sales (B) is given by: Profits =-80+120P+140B-8P2 -12B2 -4PB Solve for the profit maximizing choices for gasoline and heating oil. 12) Suppose that your sales are a function of both price (P) and advertising expenses (A) given by 0=3,000-8p+25A+2pA-.5p' -3A Solve for the combination of price and advertising that maximizes sales. We need to take the derivatives with respect to p and A9) Suppose that a busy restaurant charges $9 for its octopus appetizer. At this price, an average of 48 people order the dish each night. When it raises the price to $12, the number ordered per night falls to 42. a) Assuming that demand is linear, find the demand curve the restaurant faces. So, I know that the coefficient in from of price is 2 (a $3 increase in price lowers sales by 6) b) What price should the restaurant charge to maximize revenues? So, to maximize revenues, take the derivative with respect to price and set it equal to zero. 66-4P =0 P =16.50 0=66-2(16.50) =33 R =16.50(33) = $544.50 10) Suppose that you are a cattle rancher. You are deciding when to take your cattle to market to sell. You currently have a herd of 100 cattle. Each cow currently weighs 650 pounds and is gaining 50 pounds per month. Your feed costs are $40 per month per cow. Cattle prices are currently $8 per pound, but have been falling at the rate of $0.10 per month. If you are maximizing profits, how many month from now should you sell your cows?e) What happens to consumer surplus? = 8) Suppose that you observed the following set of data: Average Business School tuition: $30,000 Average Salary for non-MBA's: $50,000 per year Average MBA salary: $90,000 per year. The length of an MBA program is 2 years and is assumed that and MBA will have a working career of 20 years after graduation. Further, suppose that, instead of going to get an MBA, you could keep your current non-MBA job and invest what you could have used to pay for tuition, risk free, at 4% per year. a) Is this set of data consistent with market equilibrium? Explain. b) If your answer to (a) is no, how will markets adjust?Use the data above to find the parameters a,b,c, and d. b) As a check of the estimated model, solve for the equilibrium price and quantity. c) Suppose that the DEA is able to seize 100M grams of cocaine and take it off the market. What will happen to the equilibrium price and quantity? d) How will cocaine revenues for drug dealers be affected?6) Suppose that you have the following demand curve. Q=400-6P +.0051 O Represents quantity demanded, P represents price and / represents average income. You know that the current market price is $20 and average income is $20,000 a) Calculate current demand. b) Calculate the price elasticity of demand. c) Calculate the income elasticity of demand 7) Suppose that you are concerned about drug use in the US. You are interested in what the impact would be if authorities could be more effective at getting drugs off the streets. The DEA has estimated the following data: Elasticity of Demand for Cocaine: -.55 Elasticity of Supply: 1 . . Current Market Price Cocaine: $80 per gram Current Cocaine Sales (annual): 950M grams a) We are using a simply supply/demand framework