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# Geometry and Topology

### Angles/segments on dotted paper

On a sheet of dot paper or on a geoboard like the one shown, create the following: 1. Right angle 2. Acute angle 3. Obtuse angle 4. Adjacent angles 5. Parallel segments 6. Intersecting segments

### Validity of the arguments by using Euler Circles

Truth tables are related to Euler circles. Arguments in the form of Euler circles can be translated into statements using the basic connectives and the negation as follows: Let p be "The object belongs to set A." Let q be "the object belongs to set B." All A is B is equivalent to p -> q No A is B is equivalent to p -> ~q

### Finding the Focus of a Parabolic Dish

A satellite dish, is the shape of a parabola. Signals coming from a satellite strike the surface of the dish and are reflected to the focus where the receiver is located. The satellite dish has a diameter of 12 ft and a depth of 2 ft. If the diameter of the dish is halved how far from the base of the dish should the receiver

### Quaternion group presentation

Show that the quaternion group Q_2 = {+1, -1, +i, -i, +j, -j, +k, -k} has presentation <a,b|a^4 =1, a^2 = b^2, ab = ba^3> I need a rigorous proof with explanations so that I can study and understand please. I have an exam on Thursday.

### Calculating Perimeters and Areas

1. An ecology center wants to set up an experimental garden using 300m of fencing to enclose a rectangular area of 5000 . Find the dimensions of the garden. 2. A landscape architect has included a rectangular flown bed measuring 9ft by 5ft in her plans for a new building. She wants to use two colors of flowers in the bed, on

### Open Sets, Connectivity and Continuous Functions

3. a) Let M be a connected topological space and let f : M ---> R be continuous. Pick m1,m2 2 M and suppose that f(m1) < f(m2). Let x 2 R be such that f(m1) < x < f(m2). Show that there is m M with f(m) = x. (Hint: Use a connectedness argument.) b) Give R1 the usual product topology as the product of infinite copies of the rea

### Three-Dimensional Topological Group

Please help with the following problem. Let M = SL(2) be the set of 2 × 2 matrices with unit determinant. Show that, when regarded as a subset of R4 under ( a b ) ( c d ) <--> (a, b, c, d) Exists R4 and equipped with subspace topology, SL(2) becomes a 3-dimensional topological group. That is, show that (i) SL(2) is a g

### Geometry - Converses

4. State the converse of each of the following statements: (a) If lines l and m are parallel, then a transversal to lines l and m cuts out congruent alternate interior angles. (b) If the sum of the degree measures of the interior angles on one side of transversal r is less than 180°, then lines l and m meet on that side of tr

### Volume of a Conical Slice

20. R is a slice of thickness k perpendicular to the axis of a right circular cone having maximum radius b and minimum radius a. Show that its volume is V(R) = pi/3(a^2 + ab + b^2)k. Explain how Exercise 18 and 19 are essentially special cases of this. 18. C is a cone of height h and base radius a. Show V(C) = pi/3*a^2*h.

### Lines and Intersections

Incidence Geometry: Give proofs for the following: i.For every line there is at least one point not lying on it. ii.For every point, there is at least one line not passing through it iii.For every point P, there exist at least two distinct lines through P.

### Length of a rectangle is 4 centimeters more than its width

The length of a rectangle is 4 centimeters more than its width. If the width is increased by 2 centimeters and the length is increased by 3 centimeters, a new rectangle is formed having an area of 44 square centimeters more than the area of the original , rectangle. Find: 1. The dimensions of. The, new rectangle 2. The perim

### Order of operations, Area and Volume of figures

I need to know what steps are taken to come to the answer to these two problems. (8-6)^5 - 28 Evaluate. 72 ÷ 8 - 9 ÷ 3 What and how do I find the area of this figure? What and how do I find the volume of the solid shown? How do I show the answer in simplest form for the next three problems?

### Geometry of Circumference and Sphere

The equatorial radius of the earth is approximately 3960 mL. Suppose that a wire is wrapped tightly around the earth at the equator. Approximately how much must this wire be lengthened if it is to be strung all the way around the earth on poles 10 ft above ground?

### Linear function applications

Library Assignment Part 1: What is the formula for the volume of a rectangular solid? Find an object in your residence that has the shape of a rectangular solid. Measure and record the length, width, and height of your object in either centimeters (to the nearest 10th of a centimeter) or inches (to the nearest quarter of an inc

### Polygons : Angles Encountered as One Walks Around a Polygon

Draw three different nonconvex polygons. When you walk around a polygon, at each vertex you need to turn either right (clockwise) or left (counterclockwise). A turn to the left is measured by a positive number of degrees and a turn to the right by a negative number of degrees. Find the sum of the measures of the turn angles of t

### Characteristic function proof

Let A be a subset of R^n. Show that the characteristic function Xa is continuous on the interior of A and on the interior of its complement A' but is discontinuous on the boundary &#8706;A = A (bar)&#8745;A' (bar)

### Proofs that an Inclusion Map is Continuous

Show that the inclusion map i:Q -> R defined by i(q)=q for all q in Q, is continuous where both Q (rational numbers) and R(real numbers) are given the order topology.

### Mean, Median, Mode, Tables, Pictographs and Bar Graphs

Find the median of each set of numbers. 14. 1, 4, 9, 15, 25, 36 Find the mode of each set of numbers. 18. 41, 43, 56, 67, 69, 72 20. 9, 8, 10, 9, 9, 10, 8 Solve the following applications. 24. Statistics. A salesperson drove 238, 159, 87, 163, and 198 miles (mi) on a 5-day trip. What was the mean number of mil

### Vectors

Consider a regular tetrahedron with vertices: (0,0,0) , (k,k,0) , (k,0,k) , and (0,k,k) a) sketch the graph of the tetrahedron b) find the length of each edge c) find the angle between any two edges d) Find the angle between the line segments from the centroid ( k/2, k/2, k/2) to two vertices.

### Orthogonal planes.

Given two planes with equations x + 2y + z = 1 and x - 2y + 3z = 3, (a) Show that the planes are orthogonal. (b) Find the plane that is orthogonal to the given ones and passes through the point (1;-1; 2).

### Coordinate Geometry ...

(1) Show that the points A(4, 1), B(2, -3), C(0, 3) cannot form an equilateral triangle. (2) Draw the graph of the straight line 3x ?4y = 12 (3) Find the equation of a line parallel to the x- axis and passing through the point (5, 7).

### Radians, Circles, and Angular Velocity

1. Find the area of a sector having a central angle of 60° in a circle of radius 8 inches. 2. Find the perimeter and area of a circular sector whose angle is 3.5 radians if the circumference of the circle is 58 ft. 3. A point on the wheel of radius 10 feet moves with a linear velocity of 40 feet per second. Find the angul

### Metric spaces complete and incomplete

See attached Let X=(-1,1) be equipped with topology inherited from R...

### smallest possible value of an angle

T(A) is defined for A by T(A)= 3cos(A - 60) + 2 cos(A + 60) I have established that T(A) can be written SQRT(7)sin(A + 70.9), now need to find the smallest possible value of A such that T(A) + 1 = 0.

### Functional Analysis: Topological Space Proof

Just a note on notation: X*_w* is X* (set of all linear functionals) with a weak-* topology (the weakest topology in which all functionals are continuous) This posting is for #1 See attached. Let Y be a topological space...

### Finding the volume of unbounded solids of revolution

(a) Find the volume of the unbounded solid generated by rotating the unbounded region of y=e^(-x) with x>=1 around the x axis (see the attached figure) (b) What happens if y=1/sqrt(x) instead?

### GCF and area

How do you feel about mathematics now that you have completed MAT 115? Describe some coping mechanisms you developed in MAT 115 that you can use for your next math course. Example: Find the GCF of 24 and 18 Example: Calculate the area of a circle that has a radius of 8 cm (use 3.14 for pi). Example: Calculate the area o

### Golden ratio

The shape for a rectangle was one which the ratio of the length to the width was 8 to 5, the golden ratio. If the the length of a rectangular painting is 2ft longer than its width, then for what dimensions would the length and width have the golden ratio.

### Vertex Figures and Tessellations

Another definition of a regular tessellation is one whose vertex figures are identical regular polygons. A vertex figure is made by connecting the midpoints of all the edges which touch a given vertex. (1) Sketch the vertex figures for the regular semiregular tessellation of the plane and verify the definition. (2) The

### Proofs - Give an indirect proof of the theorem

Give an indirect proof of the theorem. 1. If two lines are parallel to a third line then they are parallel to each other 2. Solve the equation S=(n-2) 180 degrees for n when S is a given value. Find the number of sides of the polygon ( if possible) if the given value corresponds to the number of degrees in the sum of the in