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Geometry and Topology

Compact Sets and Compact Exhaustions

Definition: Let omega be a domain in C. Then e compact exhaustion {Ek} of omega is 1. Ek are all compact, Ek is contained in Ek+1 for all k 2. Union of Ek=omega 3. Any compact set K contained in omega is contained in some Ek Problem. Find an example of Ek's satisfying 1 and 2 but not 3 for omega=unit disk

Ratio of Surface Area to Volume

#8 How do I find the dimensions of a cube with a volume of 1000 cubic centimeters. What is the ratio of Surface Area to Volume. #9 Find the ratio of surface area to volume for a cube with volume of 64 cubic inches #10 What is the surface area of cube in exercise.

Geometry

Would like more details and explanations as to how the attached graph is solved. Create a graph with four odd vertices. See attached file for full problem description.

Geometry Word Problems, Angles, Lengths, Areas, Volumes and Graphs

Please see the attachment for the proper formatting and related diagrams. 1. DEF and GHI are complementary angles and GHI is eight times as large as DEF. Determine the measure of each angle. 2. If line p and q are parallel, find the measure of angle 2 3. (Using ABC, find the following: a) The length of s

Sets and Functions : The Symmetric Difference of Two Sets

The symmetric difference of two sets A and B, denoted by A Δ B, is defined by A Δ B = ( A - B ) U ( B - A ); it is thus the union of their differences in opposite orders. Show that A Δ ( B Δ C ) = ( A Δ B ) Δ C.

What is the volume of the solid revolution?

The region in the first quadrant bounded by the graphs of y = x and y = x^2/2 is rotated around the line y=x. Find (a) the centroid of the region and (b) the volume of the solid of revolution.

Cylindrial shells problem

Using the method of cylindrical shells to find the volume of the solid rotated about the line x=(-1) given the conditions: y=x^3 -x^2;y=0;x=0.

Circles and Cross Ratios

A) Let z1 and z2 be two points on a circle C. Let z3 and z4 be symmetric with respect to the circle. Show that the cross ratio (z1,z2,z3,z4) has absolute value 1. b)Let ad-bc=1, c not zero and consider T(z)=(az+b)/(cz+d). Show that it increases lengths and areas inside the circle|cz+d|=1 and decreases lengths and areas outsid

Projective Geometry Points and Planes

Projective Geometry Problem 1 i. Prove that a set of four points in a projective plane P (i.e. dim P = 2) form a projective frame if and only if no three of the points are collinear, i.e. no three lie on the same projective line. ii. Find a necessary and sufficient condition for five points to form a projective frame in a t

Errors in Measuring Volume of Small Balls in Large Container

Consider an experimental procedure to measure the average volume of M&M Peanut candies. One hundred piece of the candy are poured into a graduated cylinder with a 30 diameter. The cylinder is then filled with 1 mm diameter beads and shaken so that the beads and candies pack as tightly as possible. Finally, the candies are remove

Weightage Method

Four cities plan to build a new airport to serve all four communities. City B (population 180,000) is 4 miles north and 3 miles west of city A (population 75,000). City C (population 240,000) is 6 miles east and 12 miles south of city A. City D (population 105,000) is 15 miles due south of city A. Find the best location for the

Tychonoff and Hausdorff Spaces

Let X and Y connected, locally path connected and Hausdorff. let X be compact. Let f: X ---> Y be a local homeomorphism. Prove that f is a surjective covering with finite fibers. Prove: a) Any subspace of a weak Hausdorff space is weak Hausdorff. b)Any open subset U of a compactly generated space X is compactly generated

Geometry: Finding the angles of Polygons

Please help with the following problems on geometry and topology. Provide step by step calculations. See the attached files for diagrams to go along with the questions. Find the value of x and any unknown angles. Find the measure of one angle in the polygon. Round to nearest tenth if needed. 4. Regular 30-gon 5. Regular

Value of x, volume of a cylinder

1) Using the graph, what is the value of x that will produce the maximum volume? 2) The volume of a cylinder (think about the volume of a can) is given by V = πr2h where r is the radius of the cylinder and h is the height of the cylinder. Suppose the volume of the can is 121 cubic centimeters. Write h as a function

Calculating the Measurement of Height of a Cylinder

What is the measurement of the height if the radius of the cylinder is 2 centimeters? Graph this function also The formula for calculating the amount of money returned for an initial deposit money into a bank or CD is given by A=P(1+r)^nt n A is the amt of

Maximizing the Volume of an Open-Top Box

An open top box is to be constructed from a 6 by 8 foot rectangular cardboard by cutting out equal squares at each corner and then folding up the flaps. Let x denote the length of each side of the square to be cut out. Find the function V that represents the volume of the box in terms of x. Show and explain the answer, and a

Fixed Point Theorem and Closed Unit Ball in Euclidean Space

The Brouwer Fixed-Point Theorem Let denote the closed unit ball in Euclidean space : . Any continuous map from onto itself has at least one fixed point, i.e. a point such that . Proof Suppose has no fixed points, i.e. for . Define a map , , by letting be the point of intersection of and the ra

Lebesque Number and Connectivity

Lemma. Let {Ui} be an open covering of the space X having the following properties: (a) There exists a point x0 such that x0Ui for all i. (b) Each Ui is simply connected. (c) If i≠j, then Ui Uj is arcwise connected. Then X is simply connected. Prove the lemma using the following approach: To prove that any loop f: I

Writing Equations from Word Problems : Time and Distance, Two Moving Objects

A Car leaves Oak Corner at 11:33 a.m traveling south at 70km/h. at the same time, another car is 65 km west of Oak Corner traveling east at 90km/h. a. Express the distance d between the cars as a function of the time t after the first car left Oak Corner. b. show that the cars are closest to each other at noon.

Ratios and contribution analysis

Problem 1 Prepare financial analysis of Panorama. The analysis should include a summarization of the ratio analysis, explanations of what those ratios tell about the financial condition of Panorama, and a summarization of the financial strength and weaknesses of Panorama. What information do I need to use for the Financia

Volume of a solid of revolution

Find the volume of the solid generated by revolving the region described about the Y axis: Between (0,0) and (0,2), the triangular region between those points on the y-axis and the straight line x=3y/2 using the formula V=∫π[R(y)]²dy

Kilometers and starting point

Can you tell me what the answer would be if I traveled north for 35 kilometers then traveled east 65 kilometers. How far am I from my starting point? using: 35^2+65^2?

Relation in radius and height of a cylinder of given volume

Suppose the volume of a cylinder (think about the volume of a can) is given by V = πr2h where r is the radius of the cylinder and h is the height of the cylinder. Now suppose the volume of the can is 100 cubic centimeters. How do would you write h as a function of r? What is the measurement of the height if the radius o

Frieze patterns

Q 1. A frieze pattern has one and only one direction of translation. The translation isometry is denoted by T. As we have noticed in lectures the symmetry group of the fundamental pattern consists of all powers of T and is thus isomorphic to which group? Q 2. Now use Lemma F from your lecture notes to prove the following r