leper

Consider the equation xk+1 = Axk if:

A=[[2.2, 0.4] [−0.80, 1]] and x0=[5, −8]

find the eigenvalues and the eigenvalues associated with the eigenvalues

then find the solution xk of the equation.

Suppose the populations of two species in the beginning of year k, denoted by Ak and Bk, are modeled by the equations

Ak+1= 0.3Ak+1.2Bk

Bk+1= −0.4Ak+1.7Bk

If A0=18 and B0=11 initially, what will the eventual ratio of the two populations be?

The ratio Ak :Bk will eventually become 1: ______(answer)

*********************************

My attempt:

i reconized Matrix A as = [[0.3, 1.2], [-0.4, 1.7]]

thus i used det(a - llambdaI) = 0 to build a characteristic polynomial and found the eigenvalues to be

lambda = y

y1 = 1.1

y2= =0.9

checked it the trace of the matrix: 2 = y1 + y2 = 0.9 + 1.1

2.) then i found the eigenvectos of each eigenvalue

for y1 = 1.1

v1 => [[0.3 - 1.1, 1.2], [-0.4, 1.7-1.1]] -> ... [[1, -3/2], [0, 0]] => [3/2, 1] => [3,2]*

*multiplied the v1 by 2 to avoid fractions

for y2 = 0.9

v2 => [[0.3 - 0.9, 1.2], [-0.4, 1.7-0.9]] -> ... [[1, -2], [0, 0]] => [2, 1]

3.) finding c1 and c2 for equation x0 = c1v1 + c2v2

i reconized the A0=18 and B0=11 initial values as the vector x0

i reconized c1v1 + c2v2 to be the matrix [[3,2], [2,1]]

so using ax = b i could find values for c1 and c2

so using the matrix [[3,2,18],[2,1,11]] i row reduced it to [[1,0,4],[0,1,3]]

thus c1 = 4,

c2 = 3

4.) finding ak and bk to find the ratios

if Xk+1 = A*Xk

then: Xk = A^k*x0

then [ak, bk] = A^k (c1v1 + c2v2)

then [ak, bk] = (y1^k)(c1v1) + (y2^k)(c2v2)

then ak = (1.1^k)(4)[3, 2]

and bk = (0.9^k)(3)[2,1]

then ak : bk (has to be 1: some bk)

(1.1^k)(4)[3, 2] : (0.9^k)(3)[2,1]

this is all i have gotten to, i have no idea how to express the ratio correctly, or even if all my calculations to get to here is correct.

Suppose the populations of two species in the beginning of year k, denoted by Ak and Bk, are modeled by the equations

Ak+1= 0.3Ak+1.2Bk

Bk+1= −0.4Ak+1.7Bk

If A0=18 and B0=11 initially, what will the eventual ratio of the two populations be?

The ratio Ak :Bk will eventually become 1: ______(answer)

*********************************

My attempt:

i reconized Matrix A as = [[0.3, 1.2], [-0.4, 1.7]]

thus i used det(a - llambdaI) = 0 to build a characteristic polynomial and found the eigenvalues to be

lambda = y

y1 = 1.1

y2= =0.9

checked it the trace of the matrix: 2 = y1 + y2 = 0.9 + 1.1

2.) then i found the eigenvectos of each eigenvalue

for y1 = 1.1

v1 => [[0.3 - 1.1, 1.2], [-0.4, 1.7-1.1]] -> ... [[1, -3/2], [0, 0]] => [3/2, 1] => [3,2]*

*multiplied the v1 by 2 to avoid fractions

for y2 = 0.9

v2 => [[0.3 - 0.9, 1.2], [-0.4, 1.7-0.9]] -> ... [[1, -2], [0, 0]] => [2, 1]

3.) finding c1 and c2 for equation x0 = c1v1 + c2v2

i reconized the A0=18 and B0=11 initial values as the vector x0

i reconized c1v1 + c2v2 to be the matrix [[3,2], [2,1]]

so using ax = b i could find values for c1 and c2

so using the matrix [[3,2,18],[2,1,11]] i row reduced it to [[1,0,4],[0,1,3]]

thus c1 = 4,

c2 = 3

4.) finding ak and bk to find the ratios

if Xk+1 = A*Xk

then: Xk = A^k*x0

then [ak, bk] = A^k (c1v1 + c2v2)

then [ak, bk] = (y1^k)(c1v1) + (y2^k)(c2v2)

then ak = (1.1^k)(4)[3, 2]

and bk = (0.9^k)(3)[2,1]

then ak : bk (has to be 1: some bk)

(1.1^k)(4)[3, 2] : (0.9^k)(3)[2,1]

this is all i have gotten to, i have no idea how to express the ratio correctly, or even if all my calculations to get to here is correct.

2x - 5y = a

3x − 7y = b

i have no idea how to do this type of question

−8x + 8y = -20

10x + ky = -22

has a solution of k that cannot be 10 (i solved this by myself and its correct)

The solution is x =

y =

The cross-sectional area of an adjustable single loop is reduced from 1.0 m2 to 0.25 m2 in 0.10 s. what is the average emf that is induced in this coil if it is in a region where B = 2.0 T upward, and the coil's plane is perpendicular to B?

Choices a. 15 V

b. 18 V

c. 12 V

d. 21 V

My attempt:

E = - (change in flux) / (change in time)

magnetic field is constant, area is changing, time changing

flux = BA

= - (0.25 - 1)(0.2) / 0.1 = 1.5V

The answer is 15 V !! what am i doing wrong?

1.) Alpha particles from a particular radioactive source has a speed of 1.85 E7 m/s. how large of a magnetic fieldis required to bend the path of the particles into a circle 0.580 m radius. aha particle has mass 6.64E -27kg, charge 3.20R-19? ans 0.662t

This is what i did:

v = 1.85E7

m = 6.64E -27kg

q = 3.20 E-19

r = 0.580 *2

r = qm/vb

b = qm/rv

=((3.20E-18)(6.64E-27))/((0.580*2)(1.8…

= 9.90 E-53 T ???

2.) A proton with a charge 1.60E-19 and a mass of 1.67E-27kg is introduced into a region of B = 1.05T with initial velocity of 1.2E6 m/s perpendicular to B what is the radius of the path? No answer provided.

What i did:

r = qm/vb

= (1.6E-19 * 1.67e-27) / (1.05 * 1.5E6) = 2.12 E -52 ?

my answers are extremely off from the answers so i know something is definety wrong

A vertical wire carries a current straight up in a region where the magnetic field vector points due north. What is the direction of the resulting force on this current.

a)east, b west, c north d south

I tried applying the right hand rule to solve this but im not sure since if the current is straight up (represented by the index finger) then the magetic field could be either pointing into or out of the page. When the question states that the magnetic field points north does it translate to the middle finger (representing the field) sticking out towards me? In that case should the resulting force be heading north?

A thin copper rod 1.0 m long has a mass of 0.050 kg and is in a magnetic field of 0.10 T. What minimum current in the rod is needed in order for the magnetic force to cancel the weight of the rod?

a. 1.2 A

b. 2.5 A

c. 4.9 A

d. 9.8 A

in this lab i was told find the spring constant (k) and period (T) of a spring. After collecting some data I generated a force vs extension of string plot which was 7.2793x - 0.6102. I then calculated the slope to be 7.2793, is this the spring constant (K)?

in the second part of the lab, I found the average time for spring to oscillate once with the same spring with various masses of weights and was told to plot a period over square root of mass graph.

which was y = 2.0041x - 0.0112

so the slope was 2.0041 which I think is also the spring constant.

is this true then? could the spring constant be 2 different numbers or am i doing something wrong?

are the 2 slopes supposed to be the same (which i think should)

A company manufactures x units of one item and y units of another. The total cost in dollars, C, of producing these two items is approximated by the function

C = 4 x^2 + 3 xy + 3 y^2 + 900.

If the production quota for the total number of items (both types combined) is 96, find the minimum production cost.

I tried this problem numerous times throughout the week, but i simplly don't know how to isolate x ,y or lambda by itself

A company manufactures x units of one item and y units of another. The total cost in dollars, C, of producing these two items is approximated by the function

C = 4 x^2 + 3 xy + 3 y^2 + 900.

If the production quota for the total number of items (both types combined) is 96, find the minimum production cost.

The problem is i don't know how to set these questions up, help

Use Lagrange multipliers to find the maximum and minimum values of f(x,y)=4x−3y subject to the constraint x^2+2y^2 = 82, if such values exist.

i have

nabla f < 4, -3>

λg<2x, 4y>

x^2+2y^2 - 82 = 0

4 = λ2x

-3 = λ4y

im stuck at this point since i dont know how to isolate a x or y or λ in order to find the critical pont. help

i know the answer are (1/27, 1/30) by plugging in the function into a graphing calculator, but i want to know how to solve it by hand

Your monthly car payment in dollars is P=f(P0,t,r), where $P0 is the amount you borrowed, t is the number of months it takes to pay off the loan, and r percent is the interest rate.

Suppose that your bank tells you that the magnitude of ∂P/∂r is 25.

What are the units of this value? (i think it is dollars/percent)

Suppose that your bank tells you that the magnitude of ∂P/∂t is 45.

What are the units of this value? (I think its dollars/month)

Apparently one of the answers is incorrect, but i dont know which one is wrong.

The function f(x) is approximated near x=0 by the second degree Taylor polynomial P2(x)=3+x+6x2.

find f(0) , f '(0) , f "(0)

I thought the answer the answer was 18^0.5 but it is incorrect.

What i have so far,

y= −1 +90x^3−3x^5

dy/dx = -15x^2(x^2 -18)

critical points are x = 0, (18)^0.5, -(18)^0.5

2nd derivative = -60x(x^2 -9)

only -(18)^0.5 is the largest postive number here.

my second derivative was -66/(x+4)^3

there are no inflection points

how do you find the intervals of concaving up and concaving down?

my second derivative was 48(8x^2 +1) / (8x^2 -3)^3

there are no inflection points since -1/8 = x^2 gives no solution, but im not sure

how do you find the intervals of concaving up and concaving down?