A binary variable can be introduced to a mixed integer program to allow for a “threshold constraint.” A threshold constraint says that if any units are used, at least a specified minimum amount must be used. Define X as the number of students that will go on a planned field trip. The school will rent a bus only if at least 20 students plan to go on the trip. Define Y as a binary variable that equals 1 if X is nonzero, and equals 0 if X is zero (i.e., if nobody goes on the trip). If M represents a very large number, what two constraints can be added to the mixed integer program to ensure that if any students go on the field trip, at least 20 have to go?

1. Find the future value of each deposit if the account pays​ (a) simple​ interest, and​ (b) interest compounded annually.

​$1500 at 6​% for 8 years

​

2. . Tony opened a hot dog stand last April. He borrowed ​$6800 to pay for the stand and startup​ inventory, and he agreed to pay off the loan in 10 months at 6​% simple interest. Find the total amount required to repay the loan.

The total amount required to repay the loan is $__

3. For each​ deposit, find the future value​ (that is, the final amount on​ deposit) when compounding occurs​ (a) annually,​ (b) semiannually, and​ (c) quarterly.

1. Neema bought appliances costing ​$3775 at a store charging 6​% ​add-on interest. She made a $1000 down payment and agreed to monthly payments over four years. What percent of the original price tag total did the financing​ cost?

The financing cost was _____% of the original price tag total.

2. Use the​ add-on method of calculating interest to find the total interest and the monthly payment of a ​$650 loan for 14 months at 6.1​%.

The total interest is $____

3. How long​ (in years) will it take Michael Garbin to pay off a ​$7000 loan with monthly payments of ​$128.83 if the​ add-on interest rate is 6.1​%?

Michael Garbin would take ____ years to pay off a ​$7000 loan amount.

We start the week by introducing polynomials. We will learn how to identify and simplify polynomials. We will also learn how to find the greatest common factor (GCF) among them. As our knowledge of polynomials grows, we will then move on to factoring trinomials. For your first post, search online for an article or video that describes how polynomials can be used in the real world. Provide a one paragraph summary of the article or video in your own words.

You are the treasurer of the new island kingdom Polar Koordinatea where no calculators are allowed. The Queen summons you and gives you 24h to design a coin for the kingdom. After a sleepless night, you come up with two proposals: (i) The coin is the circle r = 1, having inside the rose r = cos(3θ) whose petals are plated in gold; (ii) The coin is circle r = 1, having inside the rose r = sin(5θ) whose petals are plated in gold. (a) Draw the two designs you submit to the Queen. (b) Seeing the designs, the Queen decides: “Make the one that has more gold. Bring it tomorrow!” What do you tell the Queen the next day? (c) Next day, after your answer, the Queen decides again: “You modify the design (i) as follows. Inside the coin r = 1, draw also the circle r = 1/2. The part of the rose r = cos(3θ) which is inside r = 1/2 shall be covered with platinum, the rest of the rose shall be covered with gold. And this shall be the coin of Polar Koordinatea!” Just when you are about to leave happy, the Queen says: “I would like to know by tomorrow if more platinum or more gold is needed for the new coin. Tell me please the exact difference between the two areas.” Can you answer this?

7. We wish to create a partition of the Cartesian plane. Determine if the following sets are a partition and explain why or why not.
   1. The set of all circles with radius 3 units and varying centers.
   2. The set of all circles with varying radii and centered at the point (0,0) (called the origin). Note the origin is a circle with radius zero.
   3. The set of all vertical lines.

Find the steady-state current ip(t) in an LRC-series circuit when L = 1 2 h, R = 20 Ω, C = 0.001 f, and E(t) = 100 sin(60t) + 200 cos(40t) V.

ip(t)

A) Define the words submartix and minor, and use theses to define the determinant? with example

B) example how to convert a linear system of equation and its matrix representation?

Use Laplace Transforms to solve the following IVPs . y′′+9y={2−t ,0} piecewise function 0≤t<2 , t≥2 ;y(0)=1 ,y′(0)=0

PART A)

Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 5300. The number of fish doubled in the first year.

Assuming that the size of the fish population satisfies the logistic equation

dPdt=kP(1−PK),

determine the constant k, and then solve the equation to find an expression for the size of the population after t years.
k=_______________  
P(t)=______________

How long will it take for the population to increase to 2650 (half of the carrying capacity)?
It will take ________________ years.

PART B)

Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation

dPdt=cln(KP)P

where c is a constant and K is the carrying capacity.

Solve this differential equation for c=0.2, K=3000 and initial population P0=200
P(t)=__________ .

Compute the limiting value of the size of the population.
limt→∞P(t)=_________

At what value of PP does PP grow fastest? ___________
P=__________

Q1. In a second-semester English class, 35% of the students are sophomores and the rest freshmen; 30% of the sophomores are repeating the course and 20% of the freshmen gained admission to the course by advanced placement. The professor randomly selects a student to comment on the assignment. (Round your answers to three decimal places.)

• Find the probability that the professor selects a freshman who gained advanced placement admission to the class.?

A) In a group of college students, 60 males and 80 females, 25% of the males and 45% of the females are from out of state. A student is randomly selected. Find the following probabilities. (See Example 9. Round your answers to three decimal places.)

• The person is a female.?
• The person is from out of state, given that the person is a male.?
• The person is from out of state when the person is selected from the females.?
• The person is from out of state.?
• The person is female when the selection is made from the out-of-state persons.?
  

A hardware store will run an advertising campaign using radio and newspaper. Every dollar spent on radio advertising will reach 60 people in the "Under $35,000" market, and 60 people in the "Over $35,000" market. Every dollar spent on newspaper advertising will reach 100 people in the "Under $35,000" market, and 20 people in the "Over $35,000" market. If the store wants to reach at least 210,000 people in the "Under $35,000" market and 240,000 people in the "Over $35,000" market, how much should it spend on each type of advertising to minimize the cost?

Minimum amount spent on advertising (in dollars):

Dollars spent on radio advertising

Dollars spent on newspaper advertising

Use Newton's method to find all solutions of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)

sqrt(x + 1) = x^2 − x

What does x equal?

Let A be a real n × n matrix, and suppose that every leading principal submatrix ofA of order k < n is nonsingular. Show that A has an LU-factorisation.