A 100 gallon tank initially contains 10 gallons of fresh water. At t=0 a brine solution containing 1 pound of salt per gallon is poured into the tank at the rate of 4 gal/min., while the well-stirred mixture leaves the tank at the rate of 2 gal/min. Find the amount of salt in the tank at the moment of overflow.

Enter answer as numerical decimal correct to two decimal places. pounds.

Show that, for each of the following fields, it is impossible to define an order relation ≤ that would make it an ordered field: (a) The set {0, 1} with addition and multiplication modulo 2. (b) The field of complex numbers.

Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of on-site labor required, and the profit per unit are as follows.

For the first year's production, a sum of $8,200,000 is budgeted for the building material; the number of labor-hours available for work in the factory is not to exceed 209,000 hr; and the amount of labor for on-site work is to be less than or equal to 243,000 labor-hours. Determine how many houses of each type Boise should produce to maximize its profit from this new venture.

how do you solve tan(theta)-sin(theta)=.25?

Identify the following groups:

(a) 〈a, b: a⁴ = b² = 1, bab = a³〉

(b) 〈a, b : a³ = b² = (ab)² = 1〉

(c) 〈a, b, c : abc = bca = cab〉
(d) 〈a,b : aba⁻¹ = b², bab⁻¹ = a²〉

Find the steady-state current i_p(t) in an LRC-series circuit when L = 1/2 h, R = 20 Ω, C = 0.001 f, and

E(t) = 100 sin(60t) + 300 cos(40t) V.

Can you please explain the difference between Eulerian paths and Hamilton paths as well as Eulerian circuits and Hamilton circuits. Can you determine if a graph is one of these? Can a graph be both?

You are planning on flying out of an airport on a trip. The airport parking garage charges $6 per day for the first four days, $4 per day for the next three days and $2 per day thereafter. A parking garage just outside the airport charges $5 per day and provides a free shuttle to the airport. When is it more cost-effective to park at the airport parking garage?

Your solution MUST include responses to ALL four parts. a) Understand the problem. Restate the problem in your own words. What do you know from the reading the problem? What are you looking for? What type of problem is this? What is needed in order to solve the problem?

b) Make a plan. State your plan for solving this problem. You may use words or diagrams. You might want to consider making a table, drawing a diagram, looking for a pattern, or building an equation or model.

c) Implement your plan. Once you have articulated your plan, carry out your plan. Using the information given, create mathematical model, an equation that you can use to determine when it will be more cost-effective to park at the airport parking garage. Clearly indicate how you arrived at that answer. Show your work! (*Even if using a calculator, discuss how and why you took the steps you did, not just what buttons you pushed.)

d) Look back. Is your answer reasonable? Can you find a way to check your work? Interpret your results. Remember that you have multiple representations – words, tables, graphs, and equations. Can you find another way to look at this problem that would allow you to check that your solution is correct? Interpret the answers in the context of the original application.

Find a subgroup G in symmetric (permutation) group Sn such that

(1) n = 4 and G is abelian noncyclic group

(2) n = 8 and G is dihedral group.

Scott secured a 5-year car lease at 6.20% compounded annually that required him to make payments of $881.83 at the beginning of each month. Calculate the cost of the car if he made a downpayment of $3,000.

Starting from mynewton write a function program mysymnewton that takes as its input a symbolic function f and the ordinary variables x0 and n. Let the program take the symbolic derivative f ′ , and then use subs to proceed with Newton’s method. Test it on f(x) = x 3 − 4 starting with x0 = 2. Turn in the program and a brief summary of the results

Let f be a function with domain the reals and range the reals. Assume that f has a local minimum at each point x in its domain. (This means that, for each x ∈ R, there is an E = Ex > 0 such that, whenever | x−t |< E then f(x) ≤ f(t).) Do not assume that f is differentiable, or continuous, or anything nice like that. Prove that the image of f is countable. (Hint: When I solved this problem as a student my solution was ten pages long; however, there is a one-line solution due to Michael Spivak.)

1. Must be nicely written up AS A PROOF.

a. Show that gcd(m + n, m) = gcd(m, n).

b. If n | k(n + 1), show that n | k.

c. Show that any two consecutive odd integers are relatively prime.

A body of mass 7 kg is projected vertically upward with an initial velocity 18 meters per second. The gravitational constant is g=98m/s². The air resistance is equal to k|v| where k is a constant.

Find a formula for the velocity at any time ( in terms of k ):

Find the limit of this velocity for a fixed time t₀ as the air resistance coefficient k goes to 0.

Given a scalar fifield φ(x, y, z)=3x²-yz and a vector field F(x, y, z)=3xyz²+2xy³j-x²yzk.

Find:

(i)F.∇φ.

(ii)F×∇φ.

(iii)∇(∇.F).