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# Geometry and Topology

### Maximum Volume of an Open-Top Box

A square sheet of cardboard 24 inches on a side is made into a box by cutting squares of equal size from each corner of the sheet and folding the projecting flaps into an open top box. What should be the length of the edge of any of the cutout squares to give the box maximum volume? 4 inches 4.5 inches

### Basic Graphing

46. Technology. Driving down a mountain, Tom finds that he has descended 1800 ft in elevation by the time he is 3.25 mi horizontally away from the top of the mountain. Find the slope of his descent to the nearest hundredth. Section 7.2 pp. 626-627 16,20,28 16. Find the slope of any line perpendic

### Real Life Applications of Geometry

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

### Application of laws of cosine and sin rule

At 1:00pm Jill left home traveling 45mph on a bearing of N 40 degrees W. At 1:30pm John left traveling 50mph on a bearing of S 75 degrees W. A) Illustrate the positions of Jill and John at 3:00pm. (I calculated that Jill would be 90 miles away and John would be 75 miles away) B) Find the measure of the angle between their

### Find the set of points of convergence of a given filter on an infinite set X with the cofinite topology. Prove that a space is compact if and only if every open cover has an irreducible subcover.

1. Let X be an infinite set, let T be the cofinite topology on X, and let F be the filter generated by the filter base consisting of all the cofinite subsets of X. To which points of X does F converge? 2. Let X be a space. A cover of X is called irreducible if it has no proper subcover. (a) Prove that X is compact if and o

### Sets and Sequences

2.) If " S " is the set of all "x" such that 0&#8804;x&#8804;1, what points, if any, are points of accumulation of both "S" and C(S)? 3.) Prove that any finite set is closed. 5.) Prove that, if "S" is open, each of its points is a point of accumulation of "S". 1.) Suppose "S" is a set having the number "M" as its least up

### Radical Equations and Basic Geometry

Please see the attached file for the fully formatted problems. Section 9.5 Solve each of the equations. Be sure to check your solutions. Exercise 8 Exercise 14 Exercise 30 Section 9.6 Exercise 15 Geometry. A homeowner wishes to insulate her attic with fiberglass insulation to conserve energy. The insulati

### Geometry Problems

Please explain in full detail the steps to these problems. Do not do #25 instead explain the following: 18. What is the area of a square if the length of a diagonal is 4 sq. rt. 2? 22. The floor of a room is 120 feet by 96 feet. The ceiling is 9 feet above the floor. Everything is to be painted except the floor. (Don't worr

### 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 and width of a rectangle.

An architect has designed a motel pool within a rectangular area that is fenced on three sides. If she uses 60 yards of fencing to enclose an area of 352 square yards, then what are the dimensions marked L and W? Assume L is greater than W.

### Compact Sets and Compact Exhaustions

Definition: Let omega be a domain in C. Then e compact exhaustion {Ek} of omega is 1. Ek are all compact, Ek is contained in Ek+1 for all k 2. Union of Ek=omega 3. Any compact set K contained in omega is contained in some Ek Problem. Find an example of Ek's satisfying 1 and 2 but not 3 for omega=unit disk

### Ratio of Surface Area to Volume

#8 How do I find the dimensions of a cube with a volume of 1000 cubic centimeters. What is the ratio of Surface Area to Volume. #9 Find the ratio of surface area to volume for a cube with volume of 64 cubic inches #10 What is the surface area of cube in exercise.

### Geometry Final Exam Problems

Please see the attached file for the fully formatted problems. keywords: collinearity, bisectors, angles, congruences, syllogism, perpendicular, congruent, triangles, Law of, tests, parallelorams, rhombus, circles, pyramids, squares, rectangles

### Geometry Final Exam Problems

Please see the attached file for the fully formatted problems. keywords: collinearity, bisectors, angles, congruences, syllogism, perpendicular, congruent, triangles, Law of, tests, parallelorams, rhombus, circles, pyramids, squares, rectangles

### Geometry

Would like more details and explanations as to how the attached graph is solved. Create a graph with four odd vertices. See attached file for full problem description.

### Equilibrium Position of a Foating Cylinder

Determine the equilibrium position of a cylinder of radius 3 inches, height 20 inches, and weight 5pi lb that is floating with its axis vertical in a deep pool of water of weight density 62.5 lb/ft^2. keywords: buoyant force

### Geometry : Angles, Lengths, Areas, Volumes And Graphs

1. DEF and GHI are complementary angles and GHI is eight times as large as DEF. Determine the measure of each angle. 2. If line p and q are parallel, find the measure of angle 2. 3. Using ABC, find the following: a) The length of side AC b) The perimeter of ABC c) The area of ABC 4. A diagonal wa

### Geometry Word Problems, Angles, Lengths, Areas, Volumes and Graphs

Please see the attachment for the proper formatting and related diagrams. 1. DEF and GHI are complementary angles and GHI is eight times as large as &#61648;DEF. Determine the measure of each angle. 2. If line p and q are parallel, find the measure of angle 2 3. (Using ABC, find the following: a) The length of s

### Volume of a Solid of Revolution : Method of Cylindrical Shells

R is bounded below by the x-axis and above by the curve y=2cosx, 0 <= x <= Pi/2. Find the volume of the solid generated by the revolving R around the y-axis by the methods of cylindrical shells.

### Sets and Functions : The Symmetric Difference of Two Sets

The symmetric difference of two sets A and B, denoted by A &#916; B, is defined by A &#916; B = ( A - B ) U ( B - A ); it is thus the union of their differences in opposite orders. Show that A &#916; ( B &#916; C ) = ( A &#916; B ) &#916; C.

### Finding the distance between two fire towers

A ranger in fire tower A spots a fire at a direction of 295 degrees. A ranger in fire tower B, located 45 miles at a direction of 45 degrees from tower A, spots the same fire at direction of 255 degrees. How far from tower A is the fire? From tower B? I need these example explained to me so that I might better understand the

### What is the volume of the solid revolution?

The region in the first quadrant bounded by the graphs of y = x and y = x^2/2 is rotated around the line y=x. Find (a) the centroid of the region and (b) the volume of the solid of revolution.

### Cylindrial shells problem

Using the method of cylindrical shells to find the volume of the solid rotated about the line x=(-1) given the conditions: y=x^3 -x^2;y=0;x=0.

### Find the angle subtended of a guy wire strung from the top of a pole and the ground.

A guy wire (a type of support used for example, on radio antennas) is attached to the top of a 50 foot pole and stretched to a point that is d feet from the bottom of the pole. Express the angle of inclination as a function of d.

### Geometry - Calculating the Maximum Area a goat can graze

If a goat is tied on a 50 foot lead to a corner on the outside of a rectangular barn and the barn is 20 feet by 20 feet and the goat can not get into the barn nor is the barn a grazing area, what is the maximum grazing area and show how the maximum grazing area was determined. A man has a barn that is 20 feet by 10 feet, he t

### Compute the total mass of a wire bent in a quarter circle

See attached file for full problem description. Compute the total mass of a wire bent in a quarter circle with parametric equations: x = 7 cost, y = 7 sint, where 0 < t < pi/2 and density function rho(x,y) = x^2 + y^2

### Circles and Cross Ratios

A) Let z1 and z2 be two points on a circle C. Let z3 and z4 be symmetric with respect to the circle. Show that the cross ratio (z1,z2,z3,z4) has absolute value 1. b)Let ad-bc=1, c not zero and consider T(z)=(az+b)/(cz+d). Show that it increases lengths and areas inside the circle|cz+d|=1 and decreases lengths and areas outsid

### Volume of a Solid of Revolution

Volume of a solid generated by the rotating the region formed by the graphs - y= x^2 , y =2, x = 0

### Projective Geometry Points and Planes

Projective Geometry Problem 1 i. Prove that a set of four points in a projective plane P (i.e. dim P = 2) form a projective frame if and only if no three of the points are collinear, i.e. no three lie on the same projective line. ii. Find a necessary and sufficient condition for five points to form a projective frame in a t

### Applications of div, grad, curl: Verification of the divergence theorem on two given surfaces

Verify the divergence theorem (&#8747;&#8747; (F.n) ds = &#8747;&#8747;&#8747; (grad.F) dV) for the following two cases: a. F = er r + ez z and r = i x + j y where s is the surface of the quarter cylinder of radius R and height h shown in the diagram below. b. F = er r^2 and r = i x + j y + k z where s is the surface of th