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Geometry and Topology

Homeomorphisms of a map

(See attached file for full problem description with proper symbols) --- Let and a map, given by . Let ~ be the equivalence relation on Xx[0,1] defined by and all other points are equivalent only to themselves. Show that Xx[0,1]/~ is homeomorphic to the Moebius strip. ---

Surjective continuous maps

(See attached file for full problem description with proper symbols and equation) --- Let be a surjective continuous map between topological spaces. Show that: a) If f is an identification mp, then for any pace Z and any map the composition is continuous if and only if g is continuous. b) If, for any space Z and any

Identification Map Descriptions

(See attached file for full problem description with proper symbols and equations) --- ? Let be the subspace of of all positive real numbers. Show that the map defined by is an identification map. ---

Calculating the amount of fencing required

A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters. What dimensions would require the least amount of fencing if no fencing is needed along the river?

Product space of the path connected topological space

Let X=X1 x X2 x...x Xn be the product space of the path connected topological space X1, X2, ..., Xn. Prove that the product space X is also path connected. Please see the attached file for the fully formatted problems.

Break Even Point and Capital Budgeting

2. (Payback period, net present value, profitability index, and internal rate of return calculations) You are considering a project with an initial cash outlay of $160,000 and expected free cash flows of $40,000 at the end of each year for 6 years. The required rate of return for this project is 10 percent. a.) What is the pr

Topology: Homeomorphism and Connectedness

Compact spaces and path connectedness (see attachment). --- ? Show that if is a homeomorphism between topological spaces, then X is path connected if and only if Y is path connected. Using open cover definition: 1) is a compact subset? 2) Is a compact subset? See the attached file.

Connectedness, Continuity, Image, Antipodal Point & Borsuk-Ulam

Show that, if X is a connected topological space and is continuous, then the image f(X) is an n interval. Show that, if is a continuous map, then if given a,b,c in with a < b and c between f(a) and f(b), there exists at least one with a and f(x)=c Let be a continuous map. Show that there exists a point in the circ

How Many Angles Are Formed?

1. Consider a point P When one ray is drawn from the point P. there are no angles formed. When two rays are drawn, one angle (if we only count angles with measures less than 180°) is formed. How many angles (with measures less than 180°) are formed when 100 rays are drawn from point P? Show all work.

Compact Set, Convergent Sequences and Subsequences and Accumulation Points

Prove that a set A, a subset of the real numbers, is compact if and only if every sequence {an} where an is in A for all n, has a convergent subsequence converging to a point in A. For the forward direction, I know that a compact set is closed and bounded, thus every sequence in A is bounded, and so has a convergent subsequen

Geometry

Exactly how many minutes is it before eight o'clock, if 40 minutes ago, it was three times as many minutes past four o'clock?

Filling a swimming pool

A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end. Water is being pumped into the pool at 1/4 cubic meter per minute, and there is 1 meter of water at the deep end. (a) what percent of the pool is filled? (b) At what rate is the water level rising?

Use double inegrals to find the volume of a region.

Find the volume of the following region in space: The first octant region bounded by the coordinate planes and the surfaces y=1-x^2, z=1-x^2. This question is #12 (section 9.3) in Advanced engineering mathmatics (8th ed.) by Kreyszig. This section deals with the evaluation of double integrals.

Use the cylindrical shell method to find volume of solid of revolution.

Show step by step work using cylindrical shell method to find the volume of the solid formed by revolving the given region about the y-axis. 22) the region bounded by the curve y=SQRT(x), the y-axis, and the line y=1. 24) the region bounded by the parabolas y=x^2, y=1-x^2, and y axis for x&#8805;0. 26) the region ins

Finding the Dimensions (Side) of a Cube Given Volume

A homemade loaf of bread turns out to be a perfect cube. Five slices of bread, each 0.6 in. thick, are cut from one end of the loaf. The remainder of the loaf now has a volume of 235 cu. in. What were the dimensions of the orginal loaf?

Volumes of Solids of Revolution and Sketches of Bounded Regions

Sketch the given region and then find the volume of the solid whose base is the given region and which has the property that each cross section perpendicular to the x-axis is a square. 2) the region bounded by the x-axis and the semi circle y = SQRT (16-x^2). Sketch the given region and then find the volume of the solid wh

Volume of a Tetrahedron

Find the volume of a tetrahedron with height h and base area B. Hint: B=(ab/2)sin(theta) Also, please see the attached document for the provided diagram of the tetrahedron. I already know that the answer is V=(Bh/3). I am simply looking for how my teacher came to this answer. Please show as many steps as possible so that

A thin glass pipe with the internal diameter of 3 mm was probed into a heart tissue (membrane) and air pressure was plied through the pipe to expand the membrane. Assuming its thickness to be negligible, the circular arc of the bulged membrane was to be 3.6mm (see picture below). Find the volume of the excessive space between the end of the glass pipe and piece of membrane. Need a solution in matlab.

A thin glass pipe with the internal diameter of 3 mm was probed into a heart tissue (membrane) and air pressure was plied through the pipe to expand the membrane. Assuming its thickness to be negligible, the circular arc of the bulged membrane was to be 3.6mm (see picture below). Find the volume of the excessive space between th

Topologies Functions Contained

Determine, for each of these topologies, which of the others in contains. --- (See attached file for full problem description)

Continuity proofs

Show that if {Aa} is a finite collection of sets... --- (See attached file for full problem description)

Clock problem involving angles

I am studying for a geometry test and am having trouble with a review problem at the end of the chapter. This is not homework. One problem asks the following: At 3:00, the hands of a clock form an angle of 90 degrees. To the nearest second, at what time will the hands of the clock next form a 90 degree angle? I figure t

General and Differential Topology

Find conditions under which the box topology is strictly finer than the product topology. See attached file for full problem description.

Neighborhoods

Show that the T1 axiom is equivalent to... --- (See attached file for full problem description)