1. When solving for the reactions at the supports of a truss, what equations do you use?

2. When solving for the forces in a member of a truss, how do you know you assumed the sense (direction) of the force in the incorrect direction? You will get negetavie values

3. When using the method of joints and you are analyzing an individual joint, how many and which equations of equilibrium can you apply?

4. When using the method of sections, what is the maximum number of members you can cut?

5. When solving a friction problem, what is the equation used to find the force of friction?

§Radiant Transportation company provides two types of boats –Boat A and Boat B. Each Boat A can carry 15 passengers, 10 vehicles, and 40 tons of goods. On the other hand, Each Boat B can carry 15 passengers, 30 vehicles, and 30 tons of goods. You are planning to hire boats from Radiant as you have to ship 120 passengers, 120 vehicles, and 120 tons of Goods. If each Boat A costs $2000 and each Boat B costs $3000 then how many of each one you should hire to minimize cost while you ship the required number of passengers, vehicles, and goods.

§Keep in mind that you have to hire at least 1 of each type of boat.

2. Let x be a real number, and consider the deleted neighborhood N∗(x;ε).

(a) Show that every element of N∗(x;ε) is an interior point.
(b) Determine the boundary of N∗(x;ε) and prove your answer is correct.

1. Describe how to find the complement and supplement of an angle.
2. Explain how to find the measure of the third angle in a triangle if you know the measures of the other two angles.
3. Explain the Pythagorean Theorem is used to find the length of one side of a right triangle if the lengths of the other two sides is known.
4. Explain how you can use the shadow of a tall object to measure its height.

1) Use MATLAB to solve this differential equation. ??/?? = .25? (1 − ?/4 ) - a

2) Use MATLAB to graph solution curves to this system with several different initial values. Be sure to show at least one solution curve for each of the scenarios found in ??/?? = .25? (1 − ?/4 ) - a ( let a = 0.16)

solve using both methods (Dsolve and ODE45 on matlab) please provide steps

1) y'+y=e^x

2) y'+2y= 2 sin(x)

Let R be a ring and n ∈ N. Let S = Mn(R) be the ring of n × n matrices with entries in R.

a) i) Let T be the subset of S consisting of the n × n diagonal matrices with entries in R (so that T consists of the matrices in S whose entries off the leading diagonal are zero). Show that T is a subring of S. We denote the ring T by Dn(R).

ii). Show that the subset I of S = Mn(R) consisting of those matrices A = (aij ) ∈ Mn(R) with aij = 0 whenever 1 ≤ i ≤ n and 1 ≤ j ≤ n − 1 is a left ideal of Mn(R).

b) Let R be a ring. An element a in R is called idempotent if a 2 = a.

i) Find all idempotent elements of the rings R = Z/6Z, and S = Z[i]

(ii) Show that if a is an idempotent in a ring R, then so is b = 1 − a.

(iii) Show that if R is a commutative ring, then the set of all idempotent elements of R is closed under multiplication

(iv) A ring B is a Boolean ring if a 2 = a for all a ∈ B, so that every element is idempotent. By considering (x + x) 2 show that 2a = 0 for any element a in a Boolean ring B.

v) Show that if B is a Boolean ring, then B is commutative.

vi) Show that if R is a commutative ring and a and b are idempotents, then a ⊕ b := a + b − ab 1 2 PROBLEM SHEET 4 is also an idempotent. Show that the set B = Idem(R) of all idempotents of R is a Boolian ring, where the addition is ⊕ and the multiplication is the same as in R.

vii) Let E be a set and let B the set of all subsets of E, show that B is a Booloian ring, where the ”multiplication” of two elements of B (i.e. subsets of E) is the intersection of these subsets, while the addition in B is given by X + Y = (X ∪ Y ) \ (X ∩ Y ) Here X, Y ∈ B (so, X ⊂ E, Y ⊂ E). What are the unit and zero elements of B?

if the involute is known show how to find the evolute ?

Prove via induction the following properties of Pascal’s Triangle:

•P(n,2)=(n(n-1))/2

• P(n+m+1,n) = P(n+m,n)+P(n+m−1,n−1)+P(n+m−2,n−2)+···+P(m,0) for all m ≥ 0

Theorem
two curves with the same intrinsic equation are necessary congruent
I need prove this theorem with details and thanks

Solve by variation of parameters:

A. y"−9y = 1/(1 − e^(3t))

B. y" +2y'+26y = e^-t/sin(5t)

Let P = (p1,...,pn) be a permutation of [n]. We say a number i is a fixed point of p, if pi = i.
(a) Determine the number of permutations of [6] with at most three fixed points.
(b) Determine the number of 9-derangements of [9] so that each even number is in an even position.
(c) Use the following relationship (not proven here, but relatively easy to see) for the Rencontre numbers:
Dn =(n-1)-(Dn-1 +Dn-2) (∗)
to perform an alternative proof of theorem 2.7. So, with the help of (∗), show that for all n ∈ N applies: n Dn =n! r=0 (-1)r r!
(Note: Of course, do not use Sentence 2.7 or Corollary 2.2, it is D0 = 1 and D1 = 0. Note that (∗) is also valid for n = 1 because of the factor (n - 1), no matter how we would define D-1. Then first look at the numbers An = Dn-nDn-1 (∗∗) and show that An = (-1)n is valid. Then divide both sides of (∗∗) by n! and deduce from this the assertion).

y'' - 8y = 4, y(1) = 9, y'(0) = 5

3) If you have a 40% probability of winning at a game of roulette, how many games can you expect to win after playing 30 games?

4) Calculate the variance of the problem above.

5) If Sarah rolls a 6-sided number cube, how many times can she expect to roll a 4 if she plays 18 games?