For an integer k, define f(k) = gcd(11k + 1, 7k + 3).
(a) Compute R = {f(k): k ∈ Z}.
(b) For each n ∈ R, find a set D_n such that, for every integer k, f(k) = n if and only if k ∈ D_n.

Is there any solution without using the 'mod' for b?

2. Find the volume of revolution by WASHER: ? = 2?^(1/2) ??? ? = x

Solve listed initial value problems by using the Laplace Transform:

3.      y^ll + 4y = t − 1,    y(0) = 1, y^l(0) = −1

1. Define a relation R on the integers by declaring xRy if 2x-3y is odd, the R is:

A) transitive, but not symmetric and not reflexive

B) reflexive and symmetric, but not transitive

C)not reflexive, not symmetric, and not transitive

D)reflexive, symmetric, and transitive

E)symmetric, but not transitive and not reflexive

2. Let R be equivalence relation on the integers defined by: xRy if x≅y(mod 8). which of the following numbers is an element of the equivalence class [18]?

A)-10

B)6

C)-6

D)12

determine the orthogonal bases for subspace of C^3 spanned by the given set of vectors. make sure that you use the appropriate inner product of C^3

A=[(1+i,i,2-i),(1+2i,1-i,i)

Kilgore's Deli is a small delicatessen located near a major university. Kilgore's does a large walk-in carry-out lunch business. The deli offers two luncheon chili specials, Wimpy and Dial 911. At the beginning of the day, Kilgore needs to decide how much of each special to make (he always sells out of whatever he makes). The profit on one serving of Wimpy is $0.45, on one serving of Dial 911, $0.58. Each serving of Wimpy requires 0.25 pound of beef, 0.25 cup of onions, and 5 ounces of Kilgore's special sauce. Each serving of Dial 911 requires 0.25 pound of beef, 0.4 cup of onions, 2 ounces of Kilgore's special sauce, and 5 ounces of hot sauce. Today, Kilgore has 20 pounds of beef, 15 cups of onions, 88 ounces of Kilgore's special sauce, and 60 ounces of hot sauce on hand.

1. Develop an LP model that will tell Kilgore how many servings of Wimpy and Dial 911 to make in order to maximize his profit today.
   

   Let
   	W = # of servings of Wimpy to make
   	D = # of servings of Dial 911 to make

   Max	W	+	D
   s.t.
   	W	+	D	≤		(Beef)
   	W	+	D	≤		(Onions)
   	W	+	D	≤		(Special Sauce)
   			D	≤		(Hot Sauce)
   			W, D	≥	0

2. Find an optimal solution. Round the answer for profit to the nearest cent and, if required, round the answers for W and D to one decimal place.
   
   Solution: W = , D = , Profit = $  
   
3. What is the dual value for special sauce? Round your answer to the nearest cent.
   
   Dual value for special sauce = $  
   
4. Increase the amount of special sauce available by 1 ounce. Give the new solution. Round the answer for profit to the nearest cent.
   
   Solution: W = , D = , Profit = $  
   
   Does the solution confim the answer to part (c)?
   
   

How can we use Hilbert's theorem 90 to find all Pythagorean triples?

Consider the homogeneous second order equation y′′+p(x)y′+q(x)y=0. Using the Wronskian, find functions p(x) and q(x) such that the differential equation has solutions sinx and 1+cosx. Finally, find a homogeneous third order differential equation with constant coefficients where sinx and 1+cosx are solutions.

A furniture factory has 2230 machine hours available each week in the cutting​ department, 1470 hours in the assembly​ department, and 2960 in the finishing department. Manufacturing a chair requires 0.3 hours of​ cutting, 0.5 hours of​ assembly, and 0.6 hours of finishing. A cabinet requires 0.8 hours of​ cutting, 0.3 hours of​ assembly, and 0.1 hours of finishing. A buffet requires 0.2 hours of​ cutting, 0.1 hours of​ assembly, and 0.9 hours of finishing. How many​ chairs, cabinets, and buffets should be produced in order to use all the available production​ capacity?

Please help and provide step by step so I can learn how to do this! Thank you :)

Question 1: Given a graph with length l(e) on edges, find a minimum length paths from a vertex s to V −s so that among all shortest lengths paths from s to V −s we find the ones with minimum number of edges.

Use Dijkstra's algorithm

consider the function

f(x)=3x-5/sqrt x^2+1. given f'(x)=5x+3/(x^2+1)^3/2 and f''(x)=-10x^2-9x+5/(x^2+1)^5/2

a) find the local maximum and minimum values. Justify your answer using the first or second derivative test . round your answers to the nearest tenth as needed.

b)find the intervals of concavity and any inflection points of f. Round to the nearest tenth as needed.

c)graph f(x) and label each important part (domain, x- and y- intercepts, VA/HA, CN, Increasing/decreasing, local min/max values, intervals of concavity/ inflection points of f?

8. Determine the centroid, ?(?̅,?̅,?̅), of the solid formed in the first octant bounded by ?+?−16=0 and 2?^2 −2(16−?)=0.