Two large containers A and B of the same size are filled with different fluids. The fluids in containers A and B are maintained at 0° C and 100° C, respectively. A small metal bar, whose initial temperature is 100° C, is lowered into container A. After 1 minute the temperature of the bar is 90° C. After 2 minutes (since being lowered into container A) the bar is removed and instantly transferred into the other container. After 1 minute in container B the temperature of the bar rises 10°. How long, measured from the start of the entire process, will it take the bar to reach 99.7° C? (Round your answer to two decimal places. Assume the final temperature being asked for is reached while the bar is container B.)

Find the pointwise limit f(x) of the sequence of functions fn(x) = x^n/(n+x^n) on [0, ∞). Explain why this sequence does not converge to f uniformly on [0,∞). Given a > 1, show that this sequence converges uniformly on the intervals [0, 1] and [a,∞) for any a > 1.

Solve y^4-4y"=g(t) using variation of parameters.

Verify the Duality Theorem 4.1.9 on each of the following LOPs.
a.
Max. z= −211x1 −189x2 −106x3 −175x4
s.t. 4x1 +6x2 −5x3 +3x4 ≤ 22
x1 +4x3 +9x4 ≤ 18
−3x1 +2x2 −2x3 ≤ 27
5x2 −2x4 ≤ 23
2x1 −x2 −x3 +x4 ≤ 16
7x1 +8x2 −7x3 +4x4 ≤ 12
& x1, x2, x3, x4 ≥ 0
b.
Max. z = x1 + 2x2
s.t. 17x1 + 21x2 ≤ 51
x1 − 4x2 ≤ 12
3x1 + 6x2 ≤ 14
& x1 , x2 ≥ 0

Problem 1. Determine the length, the size, and the minimal distance for each q-ary code C below. How many errors can each code correct? detect?

(a) C = { (0, 0, 0, 0, 0, 0), (0, 1, 1, 1, 1, 0), (1, 0, 0, 0, 0, 1), (1, 1, 1, 1, 1, 1) }. Here q = 2.

(b) C = { (0, 0, 0, 0, 0, 0, 0), (2, 1, 0, 2, 1, 0, 1), (2, 2, 2, 2, 2, 2, 2) }. Here q = 3.

(c) C = { (0, 1, 2, 3, 4), (1, 2, 3, 4, 0), (2, 3, 4, 0, 1), (3, 4, 0, 1, 2), (4, 0, 1, 2, 3) }. Here q = 5.

Problem 2. Assume the code from Problem 1(a) was used in transmission, and the following words were received. Decode each of these words using the nearest neighbour decoding algorithm. (The incomplete decoding version: if there is more than one nearest neighbour, declare an error.)

(a) (0, 1, 0, 0, 0, 0),

(b) (1, 1, 0, 0, 1, 1),

(c) (0, 1, 0, 1, 0, 1),

(d) (1, 1, 0, 0, 0, 0).

Show that the surfaces are tangent to each other at the given point by showing that the surfaces have the same tangent plane at this point. x2 + y2 + z2 − 20x − 16y + 2z + 115 = 0, x2 + y2 + 6z = 17, (5, 4, −4)

Both surfaces have the tangent plane of (answer )

at (5, 4, −4), therefore they are tangent to each other at this point.

Tak took out a loan for $5,000 at 4% interest. To repay the loan he must make a payment of $672.46 at the end of each year for 9 years. How much of his second payment is interest?  

The question we are interested in is this: After he has made payments for 3 years, how much will he still owe?  

Let l and m be two lines in ordinary Euclidean geometry intersecting at point O. Let A, B and C be three distinct points on l and A', B' and C' three distinct points on m (none of them equal to O). Suppose that AB' is parallel to BA', and AC' is parallel to CA'. Prove, using Pappus' theorem in the Extended Euclidean Plane, that BC' and CB' are parallel.

A group of young people were asked how many times they have been stopped and questioned by the police:

                                                Non-White            White

Mean # of Stops               4.95 .99

Varriance 4.11 6.08

Sample size 155 310

We need t-test in order to determine if Non-White and White young people are stopped by the police at the same rate. You should be sure to specify, and evaluate, both the Null and Alternative Hypotheses.

Please show work

A helicopter makes a force landing at sea. The last radio signal received at station C gives the bearing of the helicopter from C as at an N 57.1 degrees. E altitude of 423 feet. An observer at C sights the helicopter and gives <DCB as 12.1 degrees. How far will a rescue boat at A have to travel to reach any survivors at B? Apply the rules regarding the use of significant digits when determining your answer. Round your answer to the nearest ten feet.

Distance rescue boat travels = ________ ft

Convert the following to a decimal, scientific notation, and engineering notation

Decimal to scientific and engineering notation: 270, 14600, 0.000456, 0.022, 0.000000051, 66500000, 423000, 0.78

Scientific notation to decimal and engineering notation,4.5x10^4, 6.8x10^-8, 2.7x10^5, 7.22x10^-2, 9.52x10^-5, 1.89x10^2, 3.6x10^7, 8.62x10^-10

engineering notation to decimal and scientific notation,12k, 68p, 2.1G, 82T, 7.1n, 100m, 1.8k

In a particular card​ game, each player begins with a hand of 3 ​cards, and then draws 6 more. Calculate the probability that the hand will contain one pair​ (2 cards of one​ value, with the other cards of 7 different​ values).

Consider the following set of vectors in R6

S={[-9 7 -8 3 0 -5], [1 -7 3 2 -8 -8], [-6 -14 1 9 -23 -29], [11 -21 14 1 -16 -11], [8 16 -8 8 10 1], [17 -7 13 -8 8 18]

(a) (2 points) Demonstrate that S is not a basis for R6.

(b) (4 points) Let H = Span S. Find a basis for H and determine its dimension.

(c) (2 points) Determine whether v= [1,1,1,−1,−1,−1] ^T belongs in H or not.

(d) (6 points) Find a basis for R6 consists of the basis vectors of H found in part (b) and some additional linearly independent vectors.

(e) (4 points)Suppose A is a 6×6 matrix and T(x) =Ax. Show that T(H), the set of images of vectors of H, is a subspace of R6.

(f) (3 points) Show that dimT(H)≤dimH

(g) (4 points) Suppose A is invertible. Show that dimT(H) = dimH.

(h) (5 points)Suppose K is a 4 dimensional subspace of R6. Show that Hand K must have a nonzero vector in common.Hint: Start with bases for the two subspaces. If H andKonly have the trivial vector in common, then what’s a basis for the subspace H+K?