Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuation. Centerville is located at (12,0) in the xy-plane, Springfield is at (0,9), and Shelbyville is at (0,−9). The cable runs from Centerville to some point (x,0) on the x-axis where it splits into two branches going to Springfield and Shelbyville. Find the location (x,0) that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer.

To solve this problem we need to minimize the following function of

We find that f(x) has a critical number at x=

To verify that f(x) has a minimum at this critical number we compute the second derivative f''(x) and find that its value at the critical number is  , a positive number.
Thus the minimum length of cable needed is

I'm learning about the Nonlinear least squares: Levenberg-marquardt algorithms and I'm not understanding how it works. Is there a simple way to break it down with an equation to solve?

Prove that every open cover has a finite subcover implies that every sequence in S has a subsequence converging to a point of S

a.) Write down the formulas for all homomorphisms from Z10 into Z25.

b.)Write down the formulas for all homomorphisms from Z24 into Z18.

c.)Write down the formulas for all homomorphisms from Z into Z10.

d.)Extra: Define φ : C x → Rx by φ(a + bi) = a 2 + b 2 for all a + bi ∈ C x where R is the real numbers and C is the complex numbers. Show that φ is a homomorphism.

As part of a liability defence (see the Wikipedia page on Liebeck v. McDonald's for a similar case), lawyers at Tim Hortons have hired you to determine the temperature of a cup of Tim Horton's coffee when it was initially poured. However, you only have measurements of the coffee's temperature taken after it has been purchased. According to Newton's Law of Cooling, an object that is warmer than a fixed environmental temperature will cool over time according to the following relationship:

T(t)=E+(Tinit−E)e−ktT(t)=E+(Tinit−E)e−kt

where EE is the constant environmental temperature, and TT is the temperature of the object at time tt. The object has initial temperature TinitTinit.

Below you are given a data set measured from a purchased cup of coffee. The external temperature of the room is 2020 °C. The temperature of the coffee TiTi is given for several titi, where titi is the time in minutes since the coffee was poured.

Transform the solution T(t)T(t) by putting the exponential term on one side and everything else on the other and taking natural logs of both sides to get:

ln(T(t)−E)=ln(Tinit−E)−kt.ln⁡(T(t)−E)=ln⁡(Tinit−E)−kt.

Now transform the data below in the same way so that you can use linear least squares to estimate the unknown parameters TinitTinit and kk. Fit the transformed data to a line yi=b+axiyi=b+axi, i.e., find the values of aa and bb which minimize f(a,b)=∑i=1((yi)−(b+axi))2f(a,b)=∑i=1((yi)−(b+axi))2:

Use the computed coefficients aa and bb to calculate the following quantities:

What was the initial temperature TinitTinit of the coffee when it was poured?  °C
What is the time constant kk?  /min

Given the following adjacency matrix, A, for nodes a, b, c, and d, find the transitive closure of A. Is the result an equivalence relation, and why or why not?

Prove that an abelian group G of order 2000 is the direct product PxQ where P is the Sylow-2 subgroup of G, and Q the Sylow-5 subgroup of G. (So order of P=16 and order or Q=125).

1. Given matrices A and B, find AB and BA. Show all work. Are they equal, and why or why not?

5. Find an example in a media article (newspaper or magazine; cite your resource) where correlational data was incorrectly presented as if it were causal data. Explain why, for this data, there could be another reason for the data to correlate besides there being a causal relation between the two variables.  (6 points)

Let G(x,t) be the Green's function satisfying

          G"(x,t) + G(x,t) = δ(x-t)

with G(0,t) = 0 and G(pi/2) = 0, and the neccessary continuity conditions.

Here t is fixed, 0 < t < pi/2; the derivative means differentiation with respect to x, 0 < x < pi/2

Fill in the blanck:

G(x,t)=                                                   when 0<x<t

                                                             when t<x<pi/2

A department contains 10 part-time employees and 15 full-time employees. In how many ways can a committee be formed with 6 members if it must have more full-time employees than part-time employees?

a) Find the right form of the particular solution (you don't have to solve it)

y''-6y'+9y=6x^2+2-12e^3x

b) Solve using variation of parameters

y" + y = secx

c) define or explain the following

- Linear system

- Linear Homogenous system

- Linear Homogenous system with Constant coefficients

d) Solve the initial value problem using the D elimination

dx/dt = 4x - 3y

dy/dt = 6x-7y

Subject to the I.C

x(0) = 2

y(0) = -1