1. A cardboard container is being designed. The container will be a rectangular shape, divided into 12 smaller rectangular compartments. The bottom of the box must be a fixed area A, and strips of cardboard will be needed to form the walls and the dividers inside the box. In order to minimize costs, the container must be designed to minimize the length of cardboard used to form the edges and dividers.

   1. (a) Assume the box will be divided into a 12 by 1 grid. Determine the minimum length of cardboard needed to form the sides and dividers of the box.

   2. (b) Assume the box will be divided into a 6 by 2 grid. Determine the minimum length of cardboard needed to form the sides and dividers of the box.

   3. (c) Assume the box will be divided into a 4 by 3 grid. Determine the minimum length of cardboard needed to form the sides and dividers of the box.

   4. (d) Do the dimensions of the grid affect the minimum amount of cardboard needed? If so, which shape is most efficient?

2. The demand for a certain product at a particular retail location is estimated to be 64,000 items for the coming year. The product is stored cheaply in a warehouse, but it costs $0.80 per item per year to store them on location at the retailer. The retailer wishes to minimize costs by keeping fewer items in stock on location. The retailer must choose how many shipments per year to order from the warehouse. Each shipment will cost $100, regardless of the size of the shipment. Determine the number of shipments that will minimize total costs for the year. You may assume that the items are sold at a steady rate, and that each new shipment arrives exactly when the previous one runs out. (Hint: Storage costs can be determined by the cost per item per year, the storage time per shipment, and the average number of items in stock over this time period. The average number will be half the shipment size.)

For the matrix A, find (if possible) a nonsingular matrix P such that P−1AP is diagonal. (If not possible, enter IMPOSSIBLE.) A = 2 −2 3 0 3 −2 0 −1 2 P = Verify that P−1AP is a diagonal matrix with the eigenvalues on the main diagonal. P−1AP =

A measure of the adhesiveness of honey can be modeled as

f(x.y) = -125.48 + 4.26x + 4.85y -0.05x^2 - 0.14y^2

where x is the percentage of glucose and maltose and y is the percentage of moisture.

(a) calculate the absolute maximum of the adhesiveness of honey.

(b) the FDA restrictions for Grade A honey require that the combined percentages of glucose, maltose, and moisture equals 58%, what is the maximum measure of adhesiveness possible?

34. Use the first isomorphism theorem to prove that there is no homomorphism from D6 that has an image containing 3 elements

at a certain company, password must be formed 3-5 characters long and composed of the 26 letters of the alphabet, 10 digits 0-9, and 14 symbols.

A. how many passwords are possible if repetition is allowed?

B. what is the probability that a password chosen has at least 1 repeated character.

If t is any positive integer, describe a reduced residue system (RRS) modulo 2t , with explanation.

Prove or disprove: If G = (V; E) is an undirected graph where every vertex has degree at least 4 and u is in V , then there are at least 64 distinct paths in G that start at u.

Define the linear transformation S : P_n → P_n and T : P_n → P_n by S(p(x)) = p(x + 1), T(p(x)) = p'(x)

(a) Find the matrix associated with S and T with respect to the standard basis {1, x, x²} for P² .

(b) Find the matrix associated with S ◦ T(p(x)) for n = 2 and for the standard basis {1, x, x²}. Is the linear transformation S ◦ T invertible?

(c) Is S a one-to-one transformation? Is it onto? What is the kernel and range of S? What is the rank and nullity of S? Verify the Rank Theorem.

(d) Is T a one-to-one transformation? Is it onto? What is the kernel and range of T? What is the rank and nullity of T? Verify the Rank Theorem.

Often, cruise ships conduct all on-board transactions, with the exception of gambling, on cashless basis. At the end of the cruise, guests pay one bill that covers all onboard transactions. Suppose that 60 single travelers and 65 couples were surveyed as to their on-board bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Following is a summary of the bills for each group.

a.    Fill in the relative frequency for each group.

b.    Construct a histogram for the singles group. Scale the x-axis by $50 widths. Use relative frequency on the y-axis.

c.    Construct a histogram for the couples group. Scale the x-axis by $50 widths. Use relative frequency on the y-axis.

Prove that if two non-equal letters are interchanged in a ISBN code word the error will be detected (the word is no longer an ISBN code word)

Prove that the ISBN code can detect any single error ( if a letter was transmitted incorrectly the word is no longer an ISBN code word)

(Elgamal encryption): given elgamal encryption ciphersystem:

a)show how can we create a new legal encryption from two different encryptions that we don't know their decryptions

b)how can an adversary take advantage of the scheme at a) (what's written above), in order to attack a preknown encrypted text? elaborate

Prove true or false.

For each natural number n, ((n⁵/5)+(n^4/2)+(n^3/3)-(n/30)) is an integer

Incorrect Theorem. Let H be a finite set of n horses. Suppose that, for every subset S ⊂ H with |S| < n, the horses in S are all the same color. Then every horse in H is the same color.

i) Prove the theorem assuming n ≥ 3.

ii) Why aren’t all horses the same color? That is, why doesn’t your proof work for n = 2?

Prove the following using any method you like: Theorem. If A, B, C are sets, then (A ∪ B) \ C = (A \ C) ∪ (B \C) and A ∪ (B \ C) = (A ∪ B) \ (C \ A)

3. Let F : X → Y and G: Y → Z be functions.

i. If G ◦ F is injective, then F is injective.

ii. If G ◦ F is surjective, then G is surjective.

iii. If G ◦ F is constant, then F is constant or G is constant.

iv. If F is constant or G is constant, then G ◦ F is constant.