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# Geometry and Topology

### Angle of elevation

An outfielder throws a ball at a speed of 75 mph to the catcher who is 200 feet away. At what angle of elevation was the ball thrown? See attached formula.

### Finding the Volume of the Solid of Revolution

Find the volume of the solid generated by the revolution of a curve around an axis.

### Volume of solid by revolution

Find the volume of the solid formed when when the graph of the region bounded by y=e^x, x=0, x=2, and y=0 is revolved about the x-axis.

### Washer method

Use the washer method to find the volume of the solid of revolution formed by revolving the region bounded by f (x)= 3x^2 and g (x)=2x+1 about the x-axis. (Round the answer to 4 decimal places.)

### True Subsets Question

Let U be the set { 1, 2, 3}. What are the 8 subsets? If A and B are arbitrary subsets of U, there are 64 possible relations of the form "A is subset of B". Count the number of true ones.

### Mean Value Theorem and Maximizing Volume of an Open Box

The width of an open box is half its height and its surface area is 75 cm3. Find the dimensions to maximize its volume. Indicate which of the following functions satisfies the conditions of the mean value theorem on the given interval.

### Equation of Tangent Plane

Find the equation of the tangent plane to the surface xy + yz + zx = 3 at the point (1,1,1)

### Finding the Area of a Parallelogram

Find the area of the parallelogram with vertices (1,2,2), (1,3,6), (3,8,6), and (3,7,3)

### 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

### Given P is any point in the interior of the rectangle ABCD

Given P is any point in the interior of the rectangle ABCD. Show BP^2 + PD^2 = AP^2 + CP^2. Is the result the same when P in the exterior of the rectangle? keywords: vertex

### Six Geometry Word Problems

The finished product should be one combined document for the entire group, showing all calculations and graphical representations used. 1. Find the length L from point A to the top of the pole. 2. Lookout station A is 15 km west of station B. The bearing from A to a fire directly south of B is S 37°50' E. How far is 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.

### Volume of a cylinder

The volume of a cylinder (think about the volume of a can) is given by V = pi r2h 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. a) Write h as a function of r. Keep pi in the function's equation. b) What is the measurement of the height if

### 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.

### Volume of a sphere and surface area of a box

I need to find the dimensions of a box to fit a #5 size soccer ball that weighs 14 to 16 ounces and has a circumference of 27 to 28 inches. Then I need to find the amount of wrapping paper I would need to wrap the above box.

### Tangent Plane to Parametric Surface

Find an equation of the tangent plane to the parametric surface x = -1rcos(theta), y = -5rsin(theta), z = r at the point (-1sqrt(2), -5 sqrt(2), 2), when r = 2 and theta = pi/4.

### 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

### Graph of Functions: Vertical and Horizontal Asymptotes

Sketch the graph of the following functions. Find all vertical and horizontal asymptotes of the graph of each function. Determine intervals of increasing and decreasing, determine concavity, and locate all critical points and points of inflection. Show all special features such as cusps or vertical tangents. See attached file

### 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

### Finding the Central Angle of a Sector

If a sector of a given circle has an area of 875ft2 and its radius is 23ft, find the measure of the central angle in radians.

### Fractional Transformations, Cross Ratios and Conformal Mapping

1. a) Let z1,z2,z3,z4 lie on a circle. Show that z1,z3,z4 and z2,z3,z4 determine the same orientation iff (z1,z2,z,3,z4)>0 b) Let z1,z2,z3,z4 lie on a circle and be consecutive vertices of a quadrilateral. Prove that |z1-z3|*|z2-z4|=|z1-z2|*|z3-z4|+|z2-z3|*|z1-z4|

### Rectangular stage. One side of a rectangular stage is 2 meters longer than the other. If the diagonal is 10 meters , then what are the lengths of the sides?

Rectangular stage. One side of a rectangular stage is 2 meters longer than the other. If the diagonal is 10 meters , then what are the lengths of the sides?

### 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

### I need to know the length of the second circumference.

I have a circle with a circumference of 160' with an additonal circumference 12' outside of that. I need to know the length of the second circumference. Does that make sense?

### Determining Number of Calories

(See attached file for full problem description) The number of calories K needed each day by a moderately active man who weighs w kilograms, is h centimeters tall, and is a years old can be estimated by the formula K = 19.18w + 7h - 9.52a + 92.4 A) Marv is moderately active, weighs 97 kg, is 185 cm tall, and is 55 yr

### Relations of Sets and Functions

Let U be the set {1,2}. There are four subsets. List them. If A and B are arbitrary subsets, there are 16 possible relations of the form "A is subset of B". How many true ones are there?