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Geometry and Topology

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.

Tangent Plane to Parametric Surface

Find an equation of the tangent plane to the parametric surface x = -1rcos(theta), y = -5rsin(theta), z = r at the point (-1sqrt(2), -5 sqrt(2), 2), when r = 2 and theta = pi/4.

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

Fractional Transformations, Cross Ratios and Conformal Mapping

1. a) Let z1,z2,z3,z4 lie on a circle. Show that z1,z3,z4 and z2,z3,z4 determine the same orientation iff (z1,z2,z,3,z4)>0 b) Let z1,z2,z3,z4 lie on a circle and be consecutive vertices of a quadrilateral. Prove that |z1-z3|*|z2-z4|=|z1-z2|*|z3-z4|+|z2-z3|*|z1-z4|

Projective geometry problems

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

Estimating 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

Plucker Coordinates

1. Let d = (1, 2, -2), m = (-8, 5, 1) (a) Check that d and m are orthogonal. I already check that it is orthogonal. DO NOT ANSWER THIS PART. (b) Find a vector v such that d x v = m. In other words, find an affine point on the line with Plucker coordinates (1, 2, -2, -8, 5, 1). 2. Describe a general way

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

Angles in Polygons

A- find the measure of one angle in the polygon. round to nearest tenth if needed. 1- regular 30- gon 2- regular 35- gon b- sum of angle and number of sides to polygon sum of angles number of sides to polygon 5040 1800 2160 4140 c- tell whether the stateme

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

Volume

A solid whose base is the ellipse (x^2/16)+ (y^2/9)= 1 has cross sections perpendicular to the base and parallel to the minor axis are semi-ellipses of height 5. Find the volume.

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 o

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

Vectors and analytic geometry

Please find the 1) length and 2) direction (when defined) of 1) A X B and 2) B X A Please show all work, including the grids for the determinants. Thank you. A = 2i - 2j - k B = i - k

Unit square

Is it possible to partition a unit square [0, 1] X [0, 1] into two disjoint connected subsets A and B such that A and B contain opposing corners? I.e., such that A contains (0, 0) and (1, 1), and B contains (1, 0) and (0, 1)? *----0 | | | | 0----* Evidently, A and B couldn't be path-connected because a path running fr