##### Let ?1 ⃗ , ?2 ⃗ , ?3 ⃗ be three vectors from ℝ3 such no...

Let ?1 ⃗ , ?2 ⃗ , ?3 ⃗ be three vectors from ℝ3 such no two vectors are parallel, and ?3 ⃗ is not in the plane spanned by ?1 ⃗ and ?2 ⃗ . Prove that {?1 ⃗ , ?2 ⃗ , ?3 ⃗ } forms a basis for ℝ3

##### Formulate the following problem using the following steps. a. Define the decision variables. b. Specify the...

Formulate the following problem using the following steps.

a. Define the decision variables.

b. Specify the objective function.

c. Specify the constraints and simplify them so that the left hand side of each constraint only contains terms involving the decision variables.

ChemLabs uses raw materials I and II to produce two domestic cleaning solutions A and B. The daily availabilities of raw materials I and II are 150 and 145 units respectively. One unit of solution A consumes .5 unit of raw material I and .6 unit of raw material II. One unit of solution uses .5 unit of raw material I and .4 unit of raw material II. The profits per unit of solutions A and B are $8 and$10 respectively. The daily demand for solution A lies between 30 and 150 units, and that for solution B between 40 and 200 units. Formulate and solve graphically the optimal daily production amounts of A and B

##### Nepal and Tibet can both produce butter [B] or tea [T].         If Nepal allocates all...

Nepal and Tibet can both produce butter [B] or tea [T].

If Nepal allocates all resources to butter, it can produce a maximum of 500 units a year. If all its

resources are allocated to tea, it can produce a maximum of 2500 units.

If Tibet allocates all resources to butter, it can produce a maximum of 1500 units a year. If all its resources are allocated to tea, it can produce a maximum of 3000 units of tea.

[a] Political considerations initially mean that trade is not possible between Nepal and Tibet.

If Nepal produces 1250 units of tea for itself, how many units of butter does it produce for itself?

If Tibet also produced 1250 units of tea for itself, how many units of butter does it produce for itself?

What is the combined amount of tea produced by the countries when trade is not possible? The combined amount of butter?

[b] If political considerations change and trade becomes possible, which good will Nepal trade to

[c]   If Nepal and Tibet pursue complete specialization in the production of their comparative advantage products when initiating trade, what is the combined amount of tea produced by the countries?

Under complete specialization in comparative advantage, what is the combined amount of butter?

Comparing combined production prior to the possibility of trade [see [a]] with combined production available under specialization, what are the potential gains from trade as measured by additional butter and/or tea?

[d] Assuming complete specialization by both countries, identify a specific trade [i.e. an amount of tea traded for an amount of butter] that will leave both countries better off when compared to their positions in [a].

##### Solve the following initial value problem: tdy/dt+5y=5t with y(1)=8. Put the problem in standard form. Then...

Solve the following initial value problem: tdy/dt+5y=5t with y(1)=8.

Put the problem in standard form.

Then find the integrating factor, ρ(t)=

and finally find y(t)=

##### A) Solve the initial value problem: 8x−4y√(x^2+1) * dy/dx=0 y(0)=−8 y(x)= B)  Find the function y=y(x) (for...

A) Solve the initial value problem:

8x−4y√(x^2+1) * dy/dx=0

y(0)=−8

y(x)=

B)  Find the function y=y(x) (for x>0 ) which satisfies the separable differential equation

dy/dx=(10+16x)/xy^2 ; x>0

with the initial condition y(1)=2
y=

C) Find the solution to the differential equation

dy/dt=0.2(y−150)

if y=30 when t=0

y=

##### Q: Find the power series solution centered at the ordinary point x = 0 for each...

Q: Find the power series solution centered at the ordinary point x = 0 for each equation.
((x^2) + 1) y''-6y=0

##### You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into...

You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 400 eggs and 750 cups of cream. You make a profit of $3 on each quart of Creamy Vanilla and$2 on each quart of Continental Mocha. How many quarts of each flavor should you make to earn the largest profit? HINT [See Example 2.] (If an answer does not exist, enter DNE.)

##### For each n ∈ N, let f_n(x) = a_nx + b_n where a_n,b_n are sequences of...

For each n ∈ N, let f_n(x) = a_nx + b_n where a_n,b_n are sequences of real numbers. Prove that if f_n → f on [0,1] where f is a function on [0,1], then the convergence is necessarily uniform.

##### These questions are about math cryptography 1) Encrypt the plaintext "this is a secret message" using...

These questions are about math cryptography

1) Encrypt the plaintext "this is a secret message" using the affine function
f(x) = 5x + 7 mod 26.

2) Determine the number of divisors of 2n, where n is a positive integer.

##### For each of the subspaces ? and ? of ?4 (ℝ) defined below, find bases for...

For each of the subspaces ? and ? of ?4 (ℝ) defined below, find bases for ? + ? and ? ∩ ?, and verify that dim[?] + dim[? ] = dim[? + ? ] + dim[? ∩ ? ]

(a) ? = {(1, 3, 0, 1), (1, −2, 2, −2)}, ? = {(1, 0, −1, 0), (2, 1, 2, −1)}

(b) ? = {(1, 0, 1, 2), (1, 1, 0, 1)}, ? = {(0, 2, 1, 1), (2, 0, −1, 0)}

##### The resistance of blood flowing through an artery is R = C L r4 where L...

The resistance of blood flowing through an artery is

R = C

 L r4

where L and r are the length and radius of the artery and C is a positive constant. Both L and r increase during growth. Suppose

r = 0.1 mm,

L = 1 mm,

and

C = 1.

(a) Suppose the length increases 10 mm for every mm increase in radius during growth. Use a directional derivative to determine the rate at which the resistance of blood flow changes with respect to a unit of growth in the r-L plane.

Cr4​

(b) Use a directional derivative to determine how much faster the length of the artery can change relative to that of its radius before the rate of change of resistance with respect to growth will be positive.

(c) Illustrate your answers to parts (a) and (b) with a sketch of the directional derivatives on a plot of the level curves of R. (Use u for the unit change described in part (a) and v for the unit change described in part (b).)

##### There are 1000 mailboxes at a post office, numbered 1, 2, 3, …, 1000. There are...

There are 1000 mailboxes at a post office, numbered 1, 2, 3, …, 1000. There are also 1000 mailbox owners, one for each mailbox. At the start of the Mailbox Challenge, all mailboxes are closed and the owners open and close the mailboxes according to the following rules:

Owner 1 opens every mailbox.

Owner 2 closes every second mailbox; that is, lockers 2, 4, 6, 8, …, 1000.

Owner 3 changes the state of every third locker, closing it if it is open and

opening it if it is closed.

Owner n changes the state of every nth mailbox, etc.

When all the owners have taken their turns, how many mailboxes are open?