The Last Stop Boutique has a special sale of five days. Each day, Starting Monday, the price will decrease 10% of the previous day. For Example, the original price of the product is $20.00, the Monday of the sale will cost $18.00, on Tuesday it will cost $16.20, On Wednesday it will cost $14.58, on Thursday it will cost $13.12 and on Friday the price will be $11.81.

The price of the product that will going to use will be $25.00

A. Develop the algorithm and flowchart that calculates the rate of Withdrawal deduction according to the following values:

1. R = 10% 2. P = 8% 3. G=5%

An application of Newtonian Cooling is calculating the time of death of a person. When healthy, a human body has a steady temperature of 37◦C. Once a person dies, the regulatory mechanisms stop working, and the body temperature rises or falls, depending on the ambient temperature of the environment they died in.

1. The temperature of the body will decrease at a rate proportional to the difference in the current temperature and the ambient temperature. Based on this statement, show how we can model body temperature with the ODE

dT/ dt + kT = kT∞

where T is the temperature of the body, T∞ is the ambient temperate and k is constant.

2. Solve the 1st order ODE for T(t) given the body is initially 37◦C and ambient temperature is constant at 24◦C. Leave k as an unknown.

3. The body takes 2 hours to drop to 32◦C. Use this information to calculate the cooling coefficient k.

4. Plot the temperature of the body for 12 hours after death in MATLAB. Make sure both axes are clearly labelled. Comment on whether the plot is correctly modelling the temperature of the body and provide reasoning as to why (Hint: there are three observations you should be able to comment on!).

5. You find another body of a person related to the first (i.e. same k value), but this was outside where the ambient temperature can be approximated as:

T∞(t) = 6sin( πt /12 ) + 24

Solve the ODE in 1. for T(t) using this function for T∞.

Define the following on R3:
〈(a, b, c), (a′, b′, c′)〉 = 2aa′ + bb′ + 3cc′.

(a) Prove that 〈 , 〉 is an inner product on R3.
(b) Let B = {(1,1,0),(1,0,1),(0,1,1)}. Is B an orthogonal basis for R3 under the inner product defined above. If not, use the Gram-Schmidt algorithm to transform B into an orthogonal basis.

Using the inner product on 〈p, q〉 = ∫(0 to1)  p(x)q(x)dx on P2, write v as the sum of a vector in U and a

vector in U⊥, where v=x^2, U =span{x+1,9x−5}.

List and describe the types of carrying costs and ordering costs?

What is the ROP? How is it determined?

The school population for a certain school is predicted to increase by 80 students per year for the next 14 years. If the current enrollment is 800 students, what will the enrollment be after 14 ​years?

Joe's annual income has been increasing each year by the same dollar amount. The first year his income was ​$24,700​, and the 12th year his income was $37,900. In which year was his income $ 43,900?

How many terms are there in each of the following sequences?

a. 39,40,41,42...,539

b. 1,2,2²,2³,...2⁶⁰

c. 100, 200,300,400,...3000

d. 1,2,4,8,16,32,...2048

PLEASE SHOW WORK

Set and solve a linear system find a polynomial pp of degree 4 such that

p(0)=1, p(1)=1, p(2)=11, p(3)=61, and p(4)=205.

Your answer will be an expression in x.

Modifying your calculation, and without starting from scratch, find a polynomial qq of degree 4 such that q(0)=2, q(1)=3, q(2)=34, q(3)=167, and q(4)=522.

q(x) = ?

find the n-th order Taylor polynomial and the remainder for f(x)=ln(1+x) at 0

There are (m − 1)n + 1 people in a room. Show that either there are m people who mutually do not know each other, or there is a person who knows at least n others.

(a) Let n be odd and ω a primitive nth root of 1 (means that its order is n). Show this implies that −ω is a primitive 2nth root of 1. Prove the converse: Let n be odd and ω a primitive 2nth root of 1. Show −ω is a primitive nth root of 1. (b) Recall that the nth cyclotomic polynomial is defined as Φn(x) = Y gcd(k,n)=1 (x−ωk) where k ranges over 1,...,n−1 and ωk = e2πik/n is a primitive nth root of 1. Compute Φ8(x) and Φ9(x), writing them out with Z coefficients. Show your steps.

Prove that any graph where every vertex has degree at most 3 can be colored with 4 colors.

A SIS disease spreads through a population of size K = 30, 000 individuals. The average time of recovery is 10 days and the infectious contact rate is 0.2 × 10^(−4) individuals^(−1) day^(−1) . (a) The disease has reached steady-state. How many individuals are infected with the disease? (b) What is the minimum percentage reduction in the infectious contact rate that is required to eliminate the disease? (c) By implementing a raft of measures it is proposed to reduce the value of the infectious contact rate to one percent of its initial value. Will this be sufficient to eliminate the disease within twenty-eight days? (d) Is it feasible to eliminate the disease within twenty-eight days solely by reducing the value of the pairwise contact rate? (e) How may days will the ‘raft of measures’ have to be maintained if we are to eliminate the disease?

Let f(n,k) be the number of equivalence relations with k classes on set with n elements.

a) What is f(2,4)?

b) what is f(4,2)?

c) Give a combinational proof that f(n,k) = f(n-1,k-1)+k * f(n-1,k)

Find two integers x,y (if possible) such that 32x + 47y = 1. Is there more than one solution?