What is the process of obtaining a Fourier series solution for the heat equation with Dirichlet boundary conditions and an arbitrary initial condition.

We define a relation ∼ on R^2 by (x1,y1)∼(x2,y2) if and only if (y2−y1) ∈ 2Z. Show that the relation∼is an equivalence relation and describe the equivalence class of the point (0,1).

Sunshine Juices sold $16.6 million in the U.S. pomegranate juice drink market last year, but the 11.4 million customers who bought from Sunshine Juices spent a total of $39.2 million in this market. Across the whole marketplace, 75.6 million buyers contributed to total industry sales of $178.3 million of pomegranate juice drinks. Calculate usage index for Sunshine Juices' customers in the U.S. pomegranate juice drink market. Report your answer as a percent, and round your answer to the nearest tenth of a percent. Report 27.5%, for example, as "27.5".

Answer: 145.8. Please show the steps on how to get to 145.8.

Apply the Gram-Schmidt orthonormalization process to transform the given basis for Rⁿ into an orthonormal basis. Use the vectors in the order in which they are given.

B = {(−5, 0, 12), (1, 0, 5), (0, 2, 0)}

Use Newton's method to find the absolute maximum value of the function f(x) = 3x cos(x), 0 ≤ x ≤ π; correct to six decimal places.

Graph a sample space for the experiments:
In drawing 2 screws from a lot of right handed and left handed screws. let A,B,C,D mean at a least 1 right
handed, at least 1 left handed, 2 right handed, 2 left handed, respectively. Are A and B mutually exclusive?C and D?

Let A be a bounded linear operator on Hilbert space. Show that if R(A) is closed, so is R(A*)

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.