Problem 4. Suppose that A ⊂ R satisfies m₁(A) = 0, where m₁ denotes the one-dimensional Lebesque measure. Suppose f : R → R² satisfies

|f(x) − f(y)| ≤ (|x − y|)^1/2, for every x, y ∈ R.

Show that m₂(f(A)) = 0, m₂ denotes the two-dimensional Lebesque measure on R² .

Determine where each of the following function from R to R is differentiable and find the derivative function: a) f(x) =| x | b) g(x) = x | x | c) h(x) = sin x|sin x|.

Explain the computer components in detail for Raspberry Pi 4B (8GB) (i.e. CPU, Memory, and I/O). The explanation should include the following information:
i. CPU: CPU speed, types of CPU that are supported (if any), Number of CPU, Thread, CPU Cores, Cache, virtual memory, and any other related information).
ii. Memory: Maximum physical memory that can be used, Memory speed and capacity.
iii. I/O devices: List example I/O devices that are supported.

Problem 16-03

Grear Tire Company has produced a new tire with an estimated mean lifetime mileage of 36,500 miles. Management also believes that the standard deviation is 5000 miles and that tire mileage is normally distributed. To promote the new tire, Grear has offered to refund some money if the tire fails to reach 30,000 miles before the tire needs to be replaced. Specifically, for tires with a lifetime below 30,000 miles, Grear will refund a customer $1 per 100 miles short of 30,000.

1. For each tire sold, what is the expected cost of the promotion? If required, round your answer to two decimal places.
   
   
   
2. What is the probability that Grear will refund more than $50 for a tire? If required, round your answer to three decimal places.
   
   
   
3. What mileage should Grear set the promotion claim if it wants the expected cost to be $2.00? If required, round your answer to the hundreds place.
   
   miles

Problem 16-09

The Iowa Energy are scheduled to play against the Maine Red Claws in an upcoming game in the National Basketball Association Developmental League (NBA-DL). Because a player in the NBA-DL is still developing their skills, the number of points he scores in a game can vary. Assume that each player's point production can be represented as an integer uniform variable with the ranges provided in the table below.

1. Develop a spreadsheet model that simulates the points scored by each team. What is the average and standard deviation of points scored by the Iowa Energy? If required, round your answer to one decimal place.
   
   Average =
   
   Standard Deviation =
   
   What is the shape of the distribution of points scored by the Iowa Energy?
   
   
   
2. What is the average and standard deviation of points scored by the Maine Red Claws? If required, round your answer to one decimal place.
   
   Average =
   
   Standard Deviation =
   
   What is the shape of the distribution of points scored by the Maine Red Claws?
   
   
   
3. Let Point Differential = Iowa Energy points - Maine Red Claw points. What is the average point differential between the Iowa Energy and Maine Red Claws? If required, round your answer to one decimal place.
   
   
   
   What is the standard deviation in the point differential?
   
   
   
   What is the shape of the point differential distribution?
   
   
   
4. What is the probability of that the Iowa Energy scores more points than the Maine Red Claws? If required, round your answer to three decimal places.
   
   
   
5. The coach of the Iowa Energy feels that they are the underdog and is considering a "riskier" game strategy. The effect of the riskier game strategy is that the range of each Energy player's point production increases symmetrically so that the new range is [0, original upper bound + original lower bound]. For example, Energy player 1's range with the risky strategy is [0, 25]. How does the new strategy affect the average and standard deviation of the Energy point total?
   
   Average =
   
   Standard Deviation =
   
   Explain.
   
   The input in the box below will not be graded, but may be reviewed and considered by your instructor.
   
   
   
   How is the probability of the Iowa Energy scoring more points that the Maine Red Claws affected? If required, round your answer to three decimal places.
   
   Probability =
   
   Explain.
   
   The input in the box below will not be graded, but may be reviewed and considered by your instructor.
   

1. Problem 16-15 (Algorithmic)

   Strassel Investors buys real estate, develops it, and resells it for a profit. A new property is available, and Bud Strassel, the president and owner of Strassel Investors, believes if he purchases and develops this property it can then be sold for $160,000. The current property owner has asked for bids and stated that the property will be sold for the highest bid in excess of $100,000. Two competitors will be submitting bids for the property. Strassel does not know what the competitors will bid, but he assumes for planning purposes that the amount bid by each competitor will be uniformly distributed between $100,000 and $150,000.

   1. Develop a worksheet that can be used to simulate the bids made by the two competitors. Strassel is considering a bid of $130,000 for the property. Using a simulation of 1000 trials, what is the estimate of the probability Strassel will be able to obtain the property using a bid of $130,000? Round your answer to 1 decimal place. Enter your answer as a percent.
      
      %
      
   2. How much does Strassel need to bid to be assured of obtaining the property?
      
      $  
      
      What is the profit associated with this bid?
      
      $  
      
   3. Use the simulation model to compute the profit for each trial of the simulation run. With maximization of profit as Strassel’s objective, use simulation to evaluate Strassel’s bid alternatives of $130,000, $140,000, or $150,000. What is the recommended bid, and what is the expected profit?
      
      A bid of   results in the largest mean profit of $  .

Problem 16-13 (Algorithmic)

The wedding date for a couple is quickly approaching, and the wedding planner must provide the caterer an estimate of how many people will attend the reception so that the appropriate quantity of food is prepared for the buffet. The following table contains information on the number of RSVP guests for the 145 invitations. Unfortunately, the number of guests does not always correspond to the number of RSVPed guests.

Based on her experience, the wedding planner knows it is extremely rare for guests to attend a wedding if they notified that they will not be attending. Therefore, the wedding planner will assume that no one from these 50 invitations will attend. The wedding planner estimates that the each of the 25 guests planning to come solo has a 75% chance of attending alone, a 20% chance of not attending, and a 5% chance of bringing a companion. For each of the 60 RSVPs who plan to bring a companion, there is a 90% chance that she or he will attend with a companion, a 5% chance of attending solo, and a 5% chance of not attending at all. For the 10 people who have not responded, the wedding planner assumes that there is an 80% chance that each will not attend, a 15% chance each will attend alone, and a 5% chance each will attend with a companion.

1. Assist the wedding planner by constructing a spreadsheet simulation model to determine the expected number of guests who will attend the reception. Round your answer to 2 decimal places.
   
   guests
   
2. To be accommodating hosts, the couple has instructed the wedding planner to use the Monte Carlo simulation model to determine X, the minimum number of guests for which the caterer should prepare the meal, so that there is at least a 90% chance that the actual attendance is less than or equal to X. What is the best estimate for the value of X? Round your answer to the neares whole number.
   
   guests

A SIS disease spreads through a population of size K = 30, 000 individuals. The average time of recovery is 10 days and the infectious contact rate is 0.2 × 10^(−4) individuals^(−1) day^(−1) . (a) The disease has reached steady-state. How many individuals are infected with the disease? (b) What is the minimum percentage reduction in the infectious contact rate that is required to eliminate the disease? (c) By implementing a raft of measures it is proposed to reduce the value of the infectious contact rate to one percent of its initial value. Will this be sufficient to eliminate the disease within twenty-eight days? (d) Is it feasible to eliminate the disease within twenty-eight days solely by reducing the value of the pairwise contact rate? (e) How may days will the ‘raft of measures’ have to be maintained if we are to eliminate the disease?

Use the Mean-Value Theorem to prove that 30/203 < √ 103 − 10 < 3 /20 .

Let a and b be non-parallel vectors (algebraically a1b2 −a2b1 /= 0). For a vector c there are unique λ, µ real numbers such that c = λ· a+µ·b. proof?

solve step by step using power series solution, about x = -1, 4 (x+1)^2 y'' - 2(x+1)(x+3)y' + (x+4) y =0

Determine step by step power series solution about x=0 for:
4xy'' + 2y' - y = 0.

note: final answer must be in terms of coshx and sinhx

Offer one example of an IT or computer application that can be modeled as the TSP problem. This must be at least one paragraph. (something other than scheduling a bunch of jobs on a single machine)