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

In: Advanced Math

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

In: Advanced Math

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

Tibet? Prove your answer.

[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].

In: Advanced Math

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)=

In: Advanced Math

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=

In: Advanced Math

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

((x^2) + 1) y''-6y=0

In: Advanced Math

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.)

In: Advanced Math

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.

In: Advanced Math

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.

In: Advanced Math

The simplex algorithm is to continue in this manner, always performing basis exchanges which improve the objective function, until no more exchanges are possible. We conclude with an example: Buzz Buzz Buzz Coffee has on hand 1 kg of coffee grounds, 1 gallon of milk and 10 cups of sugar. They can use these to make espressos, containing 8 grams of grounds and no milk or sugar; lattes, containing 15 grams of grounds, 0.0625 gallons of milk and 0.125 cups of sugar; or caf´e cubano, containing 7.5 grams of grounds, no milk and 0.125 cups of sugar. They will be able to sell all they produce, which they will sell at prices of $2 for espressos, $4 for lattes and $5 for a caf´e cubano.

Question 10. (5 points) Let e, l and c be the number of espressos, lattes and caf´es cubanos manufactured, and let g, m and s be the amounts of grounds, milk and sugar left over when they are done. Let p be the amount of money they take in. Record the linear equations relating e, l, c, g, m, s and p.

Question 11. (15 points) Start at the point where no drinks are made (so e = l = c = 0). Exchange one of these variables, in order to increase p. Repeat the process of exchanging a basis variable to increase p until there are no exchanges which will make p larger. How many of each drink should be made?

In: Advanced Math

You are the owner of a lawn service company (LawnCo) which provides grounds and maintenance services to a range of corporate customers. Customers are expected to pay on the first of each month, in advance of receiving services. One of your corporate customers is an eldercare facility whose grounds you have maintained for many years. The customer has not paid for the last three months of services (from Oct.–Dec. 2020); nevertheless, to maintain a positive relationship, your company continued to provide mowing and weed control services to the eldercare facility during that time. Your company ceased providing services in January 2021 and found out in that same month that the eldercare facility filed for bankruptcy in September. Your company now believes that collection of the missed payments is extremely unlikely. Your company has already issued financial statements to lenders (for the period ending 12/31/20) which reflected revenue and a corresponding account receivable related to this customer of $10,000 per month for services provided to this customer. Those financial statements also reflected the company’s standard allowance (reserve) amount on receivables, of 4% of sales. In total, your company’s average monthly sales amount to $500,000.

Required:

1. Evaluate whether receipt of this information indicates you have a change in accounting estimate or whether the customer’s bankruptcy should result in this event being considered an error in previously issued financial statements.

2. Next, describe the accounting treatment (as required by the Codification) for each alternative, then support your explanations with draft journal entries.

3. Finally, briefly state which treatment appears to be more appropriate given the circumstances. If you must make any assumptions in reaching this conclusion, state these.

In: Advanced Math

E) Another pair are called missers. They are supposed to roll 3 & 7 more often than fair dice Below is the data from 16 rolls.

{ 10, 3, 5, 7, 3, 7, 8, 10, 6, 7, 7, 11, 3, 8, 2, 11 }

Can we say to a 10% that these dice do not roll 3 & 7 the way that fair dice are supposed to?

Remember there is a chance that the guy from gamblingcollectibles.com charged me $100 for a regular pair of dice.

In: Advanced Math

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)}

In: Advanced Math

The resistance of blood flowing through an artery is

R = C

L |

r^{4} |

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).)

In: Advanced Math

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?

In: Advanced Math