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Geometry and Topology

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)

Topology proofs

This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. Content: Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differentiable manifolds, tangent spaces,

General and Differential Topology

This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. Content: Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differentiable manifolds, tangent spaces,

General and Differential Topology

This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics. Content: Topological and metric spaces, continuity, subspaces, products and quotient topology, compactness and connectedness, extension theorems, topological groups, topological and differentiable manifolds, tangent spaces,

Volume of excessive space

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

Volume of Excess Space

A thin glass pipe with internal diameter of 3mm was probed into the heart tissue (membrane) and air pressure was applied... See attached file for full problem description.

Creative ways to teach surface area and volume.

You are part of a panel of parents, teachers, and administrators working to revise the geometry curriculum for the local high school. On tonight's agenda, you will be brainstorming creative ways to teach surface area and volume. The teachers are especially interested in methods which will help the students connect geometry to li

Volume of a drilled space in a sphere

I have enclosed all details on the attached file. How do I find the volume of the drilled area? Consider a sphere of radius R and circular hole 2d drilled through along axis of symmetry...

Length of an Arc and Volume of Dirt

See the attached file. A. The diagram over shows an area of a railway cutting that has failed in the form of a shallow rotational slip. Using radians as a measure of angular displacement determine the length of the failure surface AB. b. A partially completed site survey of a quadrilateral site is given below. You are

Geometry

If you answers are different from minds, please show steps? 1. Find the perimeter of a rectangle that is 12 ft. by 4 1/2 ft. ____a. 16 1/2 ft. _x__b. 33 ft. ____c. 48 1/2 ft. ____d. 54 ft. 2. Find the area of a rectangle that is 2.5 ft. by 4.6 ft. ____a. 38.28 ft ^2 ____b. 14. 2 ft ^ 2 __x_c. 11.5 ft ^2

Approximate Height Given a Change in Angle of Elevation

While traveling across flat land, you notice a mountain directly in front of you. The angle of elevation to the peak is 2.5 degrees. After you drive 17 miles closer to the mountain, the angle of elevation is 9 degrees. Approximate the height of the mountain. (Show all you work, including graphs).

Derive an mathematical equation for a beehive puzzle

In Fig.1 a beehive is enclosed in a bee cage. The bees can fly with a speed C. Each bee is flying non-stop from the hive to the wall of the cage and then back. Each time one returns to the hive, it is counted as one hit. There are hundreds and hundreds of bees in the cage and they fly out from the beehive in all directions. In s

Archimedean Property

I Let (X, T) be a space and A, B C X. Prove (a) ... (b) ... Also show that equality does not need to hold. (e) .... (d)...... .Also show that equality does not need to hold. (e) ...... (f) ... 2. Let B {(a, b], b ? R a < b} Show that B is a base for a topology U on R. The topology U is called the upper limit topology. 3.

5 Topology Questions (Including: de Morgan's Laws)

1. Prove the following de Morgan's laws: (a) ... (b) ... 2. Let A be a set. For each p E A, let Gp be a subset of A such that p C Gp C A. Then show that A = Up E A Gp. 3. Let f : X ---> Y be a function and A, B C Y. Then show that (a)... 4. Let f : X ?> Y be a function and A C X, B C V. Then show that (a) A C f-1 o f(A).