The rate at which a body cools also depends on its exposed surface area S. If S is a constant, then a modification of (2), given in Section 3.1, is

= kS(T − T_m),

where

k < 0

and T_m is a constant. Suppose that two cups A and B are filled with coffee at the same time. Initially, the temperature of the coffee is 145° F. The exposed surface area of the coffee in cup B is twice the surface area of the coffee in cup A. After 30 min the temperature of the coffee in cup A is 95° F. If

T_m = 65° F,

then what is the temperature of the coffee in cup B after 30 min? (Round your answer to two decimal places.)

Which of the following sets are rings with respect to the usual operations of addition and multiplication? If the set is a ring, is it also a field? can you explain in detail, I really do not understand how to prove.

1. Q(√2,√3)={a+b√2+c√3+d√6:a,b,c,d∈Q}

2. R={a+b3√3:a,b∈Q}

How manyn-digit binary strings have at least two 0s?

Problem 2. Use the FFT algorithm to evaluate f(x) = 8 − 4x + 2x 2 + 3x 3 − 5x 4 − 4x 5 + 2x 6 + x 7 at the eight 8th roots of unity mod 17. You may stop using recursion when evaluating a linear function (a + bx), which is easier to do directly. The eight 8th roots of unity mod 17 are 1, 2, 4, 8, 16, 15, 13, 9; it is easier to calculate with 1, 2, 4, 8, -1, -2, -4, -8. Do this by hand, and show your work.

Problem 1. We are going to multiply the two polynomials A(x) = 5 − 3x and B(x) = 4 + 2x to produce C(x) = a + bx + cx2 in three different ways. Do this by hand, and show your work.

(a) Multiply A(x) × B(x) algebraically.

(b) (i) Evaluate A and B at the three (real) roots of unity 1, i, −1. (Note that we could use any three values.)

(ii) Multiply the values at the three roots of unity to form the values of C(x) at the three roots.

(iii) Plug 1, i, −1 into C(x) = a + bx + cx2 to form three simultaneous equations with three unknowns.

(iv) Solve for a, b, c.

(c) (i) Evaluate A(x) and B(x) at the four (real) 4th roots of unity 1, i, −1, −i.

(ii) Multiply the values at the four 4th roots to form the values of C(x) at the four 4th roots.

(iii) Create the polynomial D(x) = C(1) + C(i)x + C(−1)x 2 + C(−i)x 3 .

(iv) Evaluate D(x) at the four 4th roots of unity 1, i, −1, −i. (v) Use these values to construct C(x).

Fluidyne Corporation manufactures and sells water filters. The economic forecasting unit of the company has supplied the following demand equation:                                      Points: 5

QB=    2000 -     5PB +   2.5PC   +   0.82Y   + 0.6AB

          (1200)       (1.5)       (1.2)            (0.5)      (0.2)

Where,

QB=quantity sold

PB=price per unit

PC=average unit price of competitor’s product

Y=income per household

AB=advertising expenditure

            R2= 0.86

S.E.E=5

Standard error of coefficients in parentheses (1200) (1.5) (1.2,) (0.5) , ( 0.2)

Given, PB=$50,    PC=$45,      AB=$12,500      Y=$2,000

1. Does each independent variable have a significant effect on the sales of Fluidyne water filters?
2. Interpret the coefficient of variables that are significant?
3. Interpret the coefficient of determination(R²)
4. Determine the monthly quantity demanded sold (QB) for water filter?
5. Is the quantity demanded for water filter (QB) sensitive to its own price?
6. Is water filter a luxury or necessity?

Consider the relation R on N such that xRy if and only if the sum of the digits of x and y coincide.

(i) Prove or disprove R is an equivalence relation. (ii) What are the equivalence classes of R.

Differential Equations. A 10kg weight is attached to a wall horizontally via a spring. The rest (equilibrium) length of the spring is 0.2m. The spring has stiffness coefficient k = 50. The weight drags along the ground, giving an effective damping coefficient of b = 40. We will ignore gravity and all other outside forces on the spring. If the weight is pulled to a position 0.5m away from the wall (thus +0.3m past equilibrium) and then flicked toward the wall at 2m/s. How close is the weight to the wall after 1 second? Bonus question: What is the closest the weight gets to the wall? Must use differential equations formula not physics

Determine if the following is is reflexive, symmetric, antisymmetric and transitive and why?

x relates y <-> x divides y 2 (on all positive numbers)

Compare your sample estimate to the current mean gas in the State of Ohio. Find the mean price of gasoline in the State of Ohio. You might try this GasBuddy website. Set up a null and alternative hypothesis to see if your sample for Dayton is enough to prove that the population mean gasoline price in Dayton is different than the mean price in Ohio. Test the hypotheses. Show your work. Using a significance level of 0.05, what is your conclusion?

Part 3 - Compare your sample estimate to the current mean gas in the U.S. What is the mean price of gasoline in the U.S.? You might try reading the graph on this GasBuddy website. Set up a null and alternative hypothesis to see if your sample for Dayton is enough to prove that the population mean price in Dayton is different than the mean price in the U.S. Test the hypotheses. Show your work. Using a significance level of 0.05, what is your conclusion?

For each of the following problems (even the book ones) do the following:

a. Use the x- and y- nullclines to find all the equilibrium points.

b. Compute the Jacobian matrix of the system.

c. Determine the type of each equilibrium point (if it is a hyperbolic equilibrium).

d. Plot the phase portrait in CoCalc.

1. dx/dt = x-xy-8x^2

dy/dt = -y+xy

2. dx/dt = x-y+x^2

dy/dt = x+y

3. dx/dt = 8x-4x^2-xy

dy/dt = 3y-3xy-y^2

4. dx/dt = y-x^2y

dy/dt = -x+xy^2

According to​ Kepler's first​ law, a comet should have an​ elliptic, parabolic, or hyperbolic orbit​ (with gravitational attractions from the planets​ ignored). In suitable polar​ coordinates, the position left parenthesis r comma script theta right parenthesis of a comet satisfies an equation of the form r = betaplus​e(r*cosine script theta​), where beta is a constant and e is the eccentricity of the​ orbit, with 0less than or equals less than1 for an​ ellipse, e equals1 for a​ parabola, and e greater than1 for a hyperbola. Suppose observations of a newly discovered comet provide the data below. Determine the type of​ orbit, and predict where the comet will be when script theta equals 4.1 left parenthesis radians right parenthesis. script theta 0.88 1.14 1.48 1.72 2.17 r 3.68 3.08 2.08 1.06 0.54 The comet has either a. hyperbolic b parabolic c elliptic  orbit. When script thetaequals4.1 ​(radians), the comet will be at requals nothing.

1.Find the derivative of the product between a scalar function and a vector function using the product formula.

2. Find the volume of an irregular solid using triple integration, the first integral should have at least one limit with variables.

3. Determine the moment of inertia of an irregular solid using triple integration. the first integral should have at least one limit with variables.

4. Find the angle between two lines using dot product. the two lines should not pass through zero.

5. Determine the work done (line integral) in a close path using two methods. The path should contain a curve and a line. the line should not pass through (0,0). The first method should be by using directly the formula F∙dr and the second method using Green's Theorem. Give your own vector field function F. F should be of the form <ax^myⁿ,ax^myⁿ>

6. Discuss a practical application of the cross product (vectors)