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    Geometry and Topology

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    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 for Undefined Terms

    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

    Subspace Topology: Interior, Closure, Boundary and Limit Points

    Consider the following subsets of (FUNCTION1) and (FUNCTION2). The subspaces X and Y of (SYMBOL) inherit the subspace topology. In the following cases determine the interior, the closure, the boundary and the limit points of the subsets: 1, 2 and 3 *(For complete problem, including properly cited functions and symbols, pleas

    Probability of a dart landing inside of one of three figures

    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

    Cylinder inside of a Cube : Ratio of Volumes

    A cylinder is inscribed in a cube. That cylinder has a sphere inscribed inside of it. What is the ratio of the volume of the sphere to the volume of the cube? HINT: Let X be a side of a square on the cube.

    Sphere inside of a Cube : Ratio of Volumes

    A cube has a sphere inscribed inside of it. It has another sphere circumscribed on the outside ot if (it being the cube). What is the ratio of the volume of the inside sphere to the volume of the outside sphere?

    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.

    Volume of Hyperspheres

    Finding formulas for the volume enclosed by a hypersphere in n-dimensional space. d) Use an n-tuple integral to find the volume enclosed by a hypersphere of radius r in n-dimensional space R^n. (Hint: The formulas are different for n even and n odd.)

    Related Rate Increases

    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: 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

    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 on Y.

    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: Homomorphisms

    Define f: [0, 1) →C by f (x) = e2πix. Prove that f is one-one, onto, and continuous. Find a point x ∈ [0, 1) and a neighborhood N of x in [0, 1) such that f (N) is not a neighborhood of f (x) in C. Deduce that f is not a homomorphism. See the attached file.

    Topology for Rectangle Papers

    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

    Tree in Polyhedron Figures

    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 Vertice Proofs

    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.

    Understanding Wave Diagrams: Golden Ratio

    Please help in understanding a "wave diagram". "The Power of Limits" by Gyorgy Doczi, is about the relationship of shape, music, nature etc to the golden section. A lot of the book has diagrams relating to an object showing how the forms relate to the golden section. One such diagram is a wave diagram, and connects a dinergic di

    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.

    Minimize Cost for a Cylindrical Can

    The metal used to make the top and bottom of a cylindrical can costs 4 cents/in^2, while the metal used for the sides costs 2 cents/in^2. The volume of the can is to be exactly 100 in^3. What should the dimensions of the can be to minimize the cost of making it? Could you please show all work so I can better grasp the conce

    Minimizing the Surface Area of a Box

    A closed rectangular box with volume 576 in^3 is to be made so its top (and bottom) is a rectangle whose length is twice its width. Find the dimensions of the box that will minimize its surface area. Could you please show all work so I can better grasp the concept? Thank you.