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# Geometry and Topology

### 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.

### 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?

### 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

### 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

### 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,

### 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

### 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

### Find the volume of a cube given the surface area.

A cube has a surface area of 54 square inches. If the length of each side is tripled, the what will the volume of the cube be?

### 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

### Conics Problem and Parametric Equations : Prove that Tangents of Two Ellipses are Perpendicular

The equations of two ellipses are i) 4x (squared) + 9y (squared) = 36 (ii) 2x (squared) + 3y (squared) = 30. A tangent to ellipse (i) meets the ellipse (ii) at the points P and Q. Show that the tangents at P and Q to ellipse (ii) are at right angles to one another. Please show this using parametric equations.

### 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).

### 8 Topology Questions : Hausdorff Space, Countability, Compactness and Homomorphisms

1. Show that the collection 13 {[O,c)J 0<c 1}u{(d,1]|O<d< 1) cannot be a base for the subspace topology [O, 1], where is the Euclidean topology on rt Hint: Use contradiction, i.e., first assume that B is a base for [O, 1]. 2. Let B {[a, b) x [c, d)Ia < b, : < d}. Show that B is a base for the product space (R x R, £ x £),

### 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).

### Find the Perimeter of an Octagon within a Circle

A regular octagon is inscribed in a circle of radius 15.8 cm. Find the perimeter of the octagon.

### Find the Variables and Internal Dimensions For a Studio and Patio

I am building a rectangular studio on south side of house, so that the north side of the studio will be a portion of the currrent south side of the house. The studio walls are 2 feet thick, and the studio's inside south wall is twice as long as its inside west wall. Also, I am building a semicircular patio around the st

### Real world geometry

Assume were 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 t

### Area of circle

An Indian sand painter begins his picture with a circle of dark sand. He then inscribes a square with a side length of 1 foot inside the circle. What is the area of the circle?

### Canonical Imbedding of the Normed Linear Space

Topology Suppose that Ε is a normed linear space. Let j: Ε→ Ε** be the canonical imbedding and let x** be a linear functional on Ε*. Then x*

### Transformation of Half Plane to Interior of a Circle

2. Show that when c1<0 the image of the half plane x<c1 under the transformation w = 1/z is the interior of a circle. What is the image when c1=0?

### Coordinate Geometry: Find the diameter of the largest circular pond that could fit in a triangular garden.

Find the diameter of the largest circular pond that could fit in a triangular garden with vertices at (18,54), (-27,36), and (27,-18), where a unit reprsents 1m.

### Maclaurin Series : Circles

4. Let C denote the circle |z|=1, taken counterclockwise, and following the steps below to show that: {see attachment for steps and equation} Please specify the terms that you use if necessary and clearly explain each step of your solution.

### Area of square inscribing a circle.

The area of a circle which is inscribed in a square is 169pi. What is the area of the square?

### 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

### 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?