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Geometry and Topology

Volume of Cone from Circle with Missing Sector

A circle has a radius of 5. A sector of that circle has a central angle of 120 degrees. This sector is cut out and the two radii folded together thus forming a cone. Find the volume of that cone.

Volume of Box Containing Spheres

240 spheres, each of radius 2, are placed in a box in 5 layers. There are 6 rows with 8 spheres in each row at each layer. The outside spheres are each tangent to the box and the spheres are tangent to those spheres next to them. Find the volume of the box which is between the spheres.

Area Bounded by Chord and Arc

In a circle, the arc of a chord is 2 pie square root 2. The radius of that circle is 3 square root 2. Find the area bounded by that chord and that arc.

Volume and Density

A hollow steel shaft 12 feet long has an outside diameter of 16in. the inside diameter is 9 inches. What is the weight of this shaft if the steel weighs 0.29 lbs per cubic inch?

What is the base of the rectangle?

A rectangle and a triangle are equal in area and also equal in height. If the base of the triangle is 40, what is the base of the rectangle?

Finite Axiomatic Geometry

Consider the following axiom system. The undefined terms are point, line, and on. Axioms: I. Given any two distinct points, there exactly one line on both of them. II. Given any line, there is at least one point not on it. III. Given any line, there are at least five points on it. IV. There is at least one line. Questio

Hits on a rectangular board

On a 12 by 20 rectangular board, three plane figures are drawn: a square 6 on a side , a circle with radius 4 and an equilateral triangle that is 8 on a side. If a dart is thrown that does hit the large rectangle what is theprobability that it hits inside one of the three plane figures? (Landing on the side of a figure counts

Topologies : Open Sets

? Let X:={a,b,c} be a set of three elements. A certain topology of X contains (among others) the sets {a}, {b}, and {c}. List all open sets in the topology T. ? Let X':={a,b,c,d,e} be a set of five elements. A certain topology T' on X' contains (among others) the sets {a,b,c}, {c,d} and {e}. List any other open set in T' which

Topology : Open Unit Balls

Please see the attached file for the fully formatted problems. Let , and denote the three metrics defined on . What are the open unit balls , and with respect to these three metrics? Make a sketch and describe them algebraically.

Related rates

The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height decreases at the rate of 0.2 cm/min. What is the rate of change of the volume of the cylinder, in cm^3/min, when the radius is 2 cm and the height is 3 cm? (Note: The volume of a right circular cylinder is V = p r^2h.)

Euler Path Problem : Traveler's Dilemma - Cross Bridges No More Than Once

Please see the attached file for the fully formatted problem. Traveler's Dilemma One day, travelers in a faraway land came upon a river with an island in the middle. On the other side of the island, the river continued but it formed two branches. The travalers also saw seven bridges that crossed the river in seven different

Topology : Subspace

Suppose (X,T) is a topological space. Let Y be non-empty subset of X. The the set J={intersection(Y,U) : U is in T} is called the subspace toplogy on Y. Prove that J indeed a toplogy on Y i.e., (Y,J) is a topological space.

Topological Space : Subspace

19. Let X be a topological space and let Y be a subset of X. Check that the so-called subspace topology is indeed a topology of Y. (question is also included in attachment)

Topology

14. Make a Mà¶bius strip out of a rectangle of paper and cut it along its central circle. What is the result? 15. Cut a Mà¶bius strip along the circle which lies halfway between the boundary of the strip and the central circle. Do the same for the circle which lies one-third of the way in from the boundary. What are the r

Undergrad 400 level Topology.

Find a tree in the polyhedron of figure 1.3 which contains all the vertices. Construct the dual graph Г and show that Г contains loops. (You don't have to construct the graph, but please describe it to me how it looks like.) (SEE ATTACHMENT)

Undergrad Topology - 400 Level

1. Prove that v(Г) - e(Г) = 1 for any tree T. (v :vertices and e : edges) 2. Even better, show that v(Г) - e(Г) ≤ 1 for any graph Г, with equality precisely when Г is a tree.

Topology : Connected Spaces and Explosion Point

Please see the attached file for the fully formatted problems. B6. (a) Define what it means for a topological space to be connected. (b) Suppose that A and B are subspaces of a topological space X, and that U C A fl B is open in both A and B in the relative topologies. Show that U is open in A U B in the relative topology.

Fubini Type I : Volume

Find the volume of the region in R3 bounded by z = 1 - x2, z = x2-1, y + z =1 and y= 0.