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# Geometry and Topology

### Finding the cubic yard

An engineer's plan shows a canal with a trapezoidal cross section that is 8 ft deep and 14 ft across at the bottom with walls sloping outward at an angle of 45 degrees. The canal is 620 ft long. A contractor bidding for the job estimates the cost to excavate the canal at \$1.75 per cu yd. If the contractor adds 10% profit, what s

### Measurement of height of a cylinder

b. The volume of a cylinder(think about the volume of a can)is given by v=pir^2h where r is the radius of the cylinder and h is the height of the cylinder. Suppose the volume of the can is 100 cubic centimeters, what is the measurement of the height of the cylinder is 2 centimeters? show work c. Graph this function

### Calculating the Measurement of Height of a Cylinder

What is the measurement of the height if the radius of the cylinder is 2 centimeters? Graph this function also The formula for calculating the amount of money returned for an initial deposit money into a bank or CD is given by A=P(1+r)^nt n A is the amt of

### Maximizing the Volume of an Open-Top Box

An open top box is to be constructed from a 6 by 8 foot rectangular cardboard by cutting out equal squares at each corner and then folding up the flaps. Let x denote the length of each side of the square to be cut out. Find the function V that represents the volume of the box in terms of x. Show and explain the answer, and a

### Create ways to teach surface area and volume to students. (Real-Life Example)

Creating ways to teach surface area and volume. I am especially interested in methods which will help students connect geometry to life in the real world.

### Maximizing the Area of a Rectangle Given Perimeter

John has 300 feet of lumber to frame a rectangular patio (the perimeter of a rectangle is 2 times length plus 2 time width) He wants to maximize the area of his patio (area of a rectangle is length times width). What should the dimensions of the patio be?

### Geometry and Construction : Finding the Hidden Vertex of an Angle

A farmer wishes to erect a striaght fence that will bisect the angle formed by two existing (straight) fences. Unfortunetly, the vertex of the angle is in the middle of a lake. How can we locate the fence line using straight edge and compass? Please see the attached file for the fully formatted problem.

### Finding length of a side of a cube given its volume.

The volume of a cube is given by V = s3. Find the length of a side of a cube if the Volume is 729 cm3.

### Word Problem

John has 300 feet of lumber to frame a rectangular patio (the perimeter of a rectangle is 2 times length plus 2 times width). He wants to maximize the area of his patio (area of a rectangle is length times width). What should the dimensions of the patio be?

### Problem Set

1. Consider the graph of y = tan x. (a) How does it show that the tangent of 90 degrees is undefined? (b) What are other undefined x values? (c) What is the value of the tangent of angles that are close to 90 degrees (say 89.9 degrees and 90.01 degrees)? (d) How does the graph show this? 2. A nautical mile depends on l

### Fixed Point Theorem and Closed Unit Ball in Euclidean Space

The Brouwer Fixed-Point Theorem Let denote the closed unit ball in Euclidean space : . Any continuous map from onto itself has at least one fixed point, i.e. a point such that . Proof Suppose has no fixed points, i.e. for . Define a map , , by letting be the point of intersection of and the ra

### Lebesque Number and Connectivity

Lemma. Let {Ui} be an open covering of the space X having the following properties: (a) There exists a point x0 such that x0Ui for all i. (b) Each Ui is simply connected. (c) If i≠j, then Ui Uj is arcwise connected. Then X is simply connected. Prove the lemma using the following approach: To prove that any loop f: I

### Writing Equations from Word Problems : Time and Distance, Two Moving Objects

A Car leaves Oak Corner at 11:33 a.m traveling south at 70km/h. at the same time, another car is 65 km west of Oak Corner traveling east at 90km/h. a. Express the distance d between the cars as a function of the time t after the first car left Oak Corner. b. show that the cars are closest to each other at noon.

### Word Problems: Volume and Radius of a Sphere

A balloon is inflated in such a way that is volume increases at a rate of 20cm^3/s. a. If the volume of the balloon was 100cm^3 when the process of inflation began what will the volume be after t seconds of inflation b. Assuming that the balloon is spherical while it is being inflated, express the radius r of the balloon a

### Word Problems: Circumference and Area of Circles

A stone is thrown into a lake, and t seconds after the splash the diameter of the circle of ripples is t meters A. Express the circumference C of this circle as a function of t. B. express the area A of this circle as a function of t.

### Ratios and contribution analysis

Problem 1 Prepare financial analysis of Panorama. The analysis should include a summarization of the ratio analysis, explanations of what those ratios tell about the financial condition of Panorama, and a summarization of the financial strength and weaknesses of Panorama. What information do I need to use for the Financia

### Word Problem : Circumference, Revolutions and Distance Traveled

The wheels of a car have a 24-in. diameter. When the car is being driven so that the wheels make 10 revolutions per second, how far will the car travel in one minute?

### Washer and shell methods

The region in the first quadrant that is bounded above by the curve y=1/&#8730;x, on the left by the line x=1/4, and below by the line y=1, is revolved about the y-axis to generate a solid. Find the volume of the solid by (a) the washer method, and (b) the shell method. Please be detailed in your response.

### Shell method to find volume about y-axis

Use the shell method formula to find the volume of the solid generated by revolving the shaded region bounded by the curves and lines below about the y-axis: V=&#8747;2&#960;(shell radius)(shell height)dx = &#8747;2&#960;x f(x)dx a&#8804;x&#8804;b Shaded region:: y=2x y=x/2 x=1

### Volume of a solid of revolution

Find the volume of the solid generated by revolving the region described about the Y axis: Between (0,0) and (0,2), the triangular region between those points on the y-axis and the straight line x=3y/2 using the formula V=&#8747;&#960;[R(y)]²dy

### Kilometers and starting point

Can you tell me what the answer would be if I traveled north for 35 kilometers then traveled east 65 kilometers. How far am I from my starting point? using: 35^2+65^2?

### 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

### Relation in radius and height of a cylinder of given volume

Suppose the volume of a cylinder (think about the volume of a can) is given by V = &#960;r2h where r is the radius of the cylinder and h is the height of the cylinder. Now suppose the volume of the can is 100 cubic centimeters. How do would you write h as a function of r? What is the measurement of the height if the radius o

### Frieze patterns

Q 1. A frieze pattern has one and only one direction of translation. The translation isometry is denoted by T. As we have noticed in lectures the symmetry group of the fundamental pattern consists of all powers of T and is thus isomorphic to which group? Q 2. Now use Lemma F from your lecture notes to prove the following r

### 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