Other Math Answers

1. Discuss ways in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry.) For each new numbering plan you discuss, show how to find the number of different telephone numbers it supports.
2. Describe at least one way to generate all the partitions of a positive integer n. (You can get idea from Exercise 49 in Section 5.3.)
3. Proof that an undirected graph has an even number of vertices of odd degree.
4. In Exercises i) and ii) determine whether the given graph has a Hamilton circuit. If it does, find such a circuit. If it does not, give an argument to show why no such circuit exists.

i)                    

ii)                  

5. Suppose that G is a connected multigraph with 2k vertices of odd degree. Show that there exist k subgraphs that have G as their union, where each of these subgraphs has a Euler path and where no two of these subgraphs have an edge in common.

Find thw volume of the solid formed by revolving the region bounded by graph(s) of the equation(s) about the x-axis.

1.y=4-x², y=0

2.y=-x+1,y=0,x=0

3.y=1/x - 1/2, y=-1/2 x +1

4.y=x²,y=4x-x²

Find the volume of the solid created by revolving the given region around the given axis.

1.y=√4-x² in quadrant I around the x-axis

2.y=x²,x=0,x=1 around the x-axis

Find the volume of the solid genrated by revolving the region bounded by the curves about the x-axis.

1.y=√9-x²,y=0

2.y=3+x , y=1+x²

3.y=sec x, y=√2, -3.14/4≤x≤3.14/4

4.y=1,y=x,x=0

Find the volume of the solid genrated by revolving the region bounded by the curves about the y-axis.

1. y=x³,x=0,y=1

2. x=y², y=x-2

3.x=csc y,y=3.14/4,y=3(3.14)/4,x=0

4.y=0,y=√x,x=4

Let S be the closed parabolic bowl consisting of two pieces: S1: z = x2 + y2

, x2 + y2 ≤ 1;

and S2: x = rCos(ɸ), y = rSin(ɸ), z = 1 for 0≤ɸ≤2π, and 0≤r≤1. Let (the vector)

F = (x-y+z)î + 2xĵ + k̂

. Compute both sides of Gauss’s Divergence Formula and show that they are equal.

Mike Marquez purchased a lawn tractor for Php 9,000. He made a down payment of Php 1,000 and financed the rest at 7% payable in 24 months. He paid off the loan at the end of the fifteenth months. Find the amount of his refund using the rule of 78s.

A thin rod of length, 𝐿 = 40 𝑐𝑚, was made from a material with a thermal 

diffusivity, 𝑘 = 2.5 𝑐𝑚²⁄𝑠 . The temperature distribution in terms of the time, 

𝑡 and the position, 𝑥 is denoted by 𝑇(𝑡, 𝑥). The following initial and boundary 

conditions are considered:

𝑇(0, 𝑥) = 𝑓(𝑥) 

𝑇(𝑡, 0) = 𝑓(0) 

𝑇(𝑡, 40) = 𝑓(40) where 𝑓(𝑥) has the following piecewise function form 

𝑓(𝑥) = { 𝑔(𝑥), 0 ≤ 𝑥 < 𝑎 

ℎ(𝑥), 𝑎 ≤ 𝑥 ≤ 40 

.

The functions 𝑔(𝑥) and ℎ(𝑥) are not constant and 𝑓(𝑥) satisfies the following 

condition, 

𝑓(40) > 𝑓(0) > 0 or 𝑓(0) > 𝑓(40) > 0.

By using a suitable function for 𝑓(𝑥) and, the values of ∆𝑥 = 4 𝑐𝑚 and ∆𝑡 = 

4 𝑠, consider TWO (2) finite-difference methods to compute the temperature 

distribution 𝑇(𝑡, 𝑥) over the time interval [0, 8].

A metal bar with an initial temperature, 𝑇₀, in the interval of 30°C ≤ 𝑇₀ ≤ 35°C

is dropped into a container of boiling water (100°C). The temperature of the 

metal bar, 𝑇 at any time, 𝑡 satisfies the following Newton’s Law of Cooling 

model

𝑑𝑇/𝑑𝑡 = −𝑘(𝑇 − 𝑇_𝑚)

where 𝑇_𝑚 is the ambient temperature and 𝑘 is the constant. After 5 seconds, 

the temperature of the bar, 𝑇₁ is in the interval of 40°C ≤ 𝑇₁ ≤ 50°C.

Find the equation that models the temperature of the metal bar, 𝑇 at 

any time, 𝑡 (choose a value of 𝑇₀ and 𝑇₁ from the given intervals, 

respectively). By using an appropriate analytical method, solve the 

derived model and explain the reason for the selection of the method.

b. Compute the temperature of the metal bar after 100 seconds by using 

the derived model in Part 1(a) with THREE (3) different numerical 

methods with step size, ℎ = 10 seconds. Select the best numerical 

method to compute the temperature of the metal bar and justify your 

answer.

reduce the Quadratic form ^x2+^y2+^z2-2xy-2yz-2zx to canonical form through an orthogonal transformtion also find its nature rank signature and index

1. Discuss the different methods of radio wave propagation.

2. Discuss the factors/parameters is designing antenna systems.

A diet is to include at least 140 milligrams of Vitamin A and at least 145 milligrams of Vitamin B. These requirements can be obtained from two types of food. Type X contains 10 milligrams of Vitamin A and 20 milligrams of Vitamin B per pound. Type Y contains 30 milligrams of Vitamin A and 15 milligrams of Vitamin B per pound. If type X food costs $12 per pound and type Y food costs $8 per pound how many pounds of each type of food should be purchased to satisfy the requirements at the minimum cost? What is the objective function?