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Geometry and Topology

Statement: If two lines are parallel then they do not intersect.

I have attached a file regarding contrapositives and inverse and converse statements --- Statement: If two lines are parallel then they do not intersect. True Converse: If 2 lines do not intersect then they are parallel False Inverse: If 2 lines are not parallel then they intersect

Al metal reacts with HCl produces AlCl3 and hydrogen gas.

Let C_0 be the space of functions f:R --> R such that lim f(x) = 0 as x goes to infinity and negative infinity C_0 becomes a metric space with sup-norm ||f|| = sup { |f(x)| : x in R } Prove that if A is a family of functions in C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions

The volume of the cross vault: Intersecting cylinders.

Let two long circular cylinders, of diameter D, intersect in such a way that their symmetry axes meet perpendicularly. Let each of these axes be horizontal, and consider the "room" above the plane that contains these axes, common to both cylinders. (In architecture this room is called a "cross vault".) The floor of the cross vau

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?

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

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.

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

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).

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 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?