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# Matrices and angles

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Project 3

2. Lance Armstrong won the 2003 Tour de France. The wheel on his bike had a 63 inch diameter. His average speed was 40 km / hour. What was the angular speed of the wheel in revolutions per hour?

3. A construction company is making picnic pavilions where the roof will be supported by two sets of beams. The beams in the front are 8 feet high and the back beams are 6.5 feet high. The distance between the front and back beams is 8 feet. At what angle will the roof lay on the front beam?

4. Latitude is the north-south location on the earth between the equator and the poles. Since the earth flattens slightly at the poles, a nautical mile varies with latitude. A nautical mile is given by N(&#61541;) = 6066 - 31 * cosine 2&#61541;&#61486;&#61472;&#61541; represents the latitude in degrees.
a) What is the length of a British nautical mile at Chicago (latitude of 42 degrees)?
b) What is the length of a British nautical mile at the north pole (latitude of 90 degrees)?
c ) At what latitude north is the length of a British nautical mile found to be 6040 feet?

5. A guy wire 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.

Be sure to demonstrate your work.

Project 3A

1. Two cars with new tires are driven at an average speed of 60mph for a test drive of 2000 miles. The diameter of the wheels of one car is 15 inches. The diameter of the wheels of the other cars is 16 inches. If the tires are equally durable and differ only by diameter, which car will probably need new tires first? Why?

2. Explain why tan(x + 450degrees) cannot be simplified using the tangent sum formulas but can be simplified using the sine and cosine formulas.
What is the difference between a trig equation that is an identity and a trig equation that is not an identity? Provide an example to clarify.

Project 4

1. 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?

2. A sail that is in the shape of an isosceles triangle has a vertex angle of 38 degrees. The angle is included by two sides, each measuring 20 ft. Find the area of the sail.
3. A shipping crate that weights 450 kilograms is placed on a loading ramp that makes an angle of 30 degrees with the horizontal. Find the magnitude of the components of the crate's weight perpendicular and parallel to the incline.

4. On an 18-hole golf course, there are par-3, par-4 and par-5 holes. A golfer who shoots par on every hole has a score of 72. The sum of the number of par-3 holes and the number of par-5 holes is 8. How many holes of each type are there on the golf course? (Remember, your total par is 72. Model par-3 holes as 3x, par-4 holes as 4x, and par-5 holes as 5x. You should have two equations and be able to solve.)
5. One slice of cheese pizza contains 290 calories, 15g of protein, 9g of fat, and 39g of carbohydrates. One-half cup of gelatin dessert contains 70 calories, 2g of protein, 0g of fat, and 17g of carbohydrates. One cup of whole milk contains 150 calories, 8g of protein, 8g of fat, and 11g of carbohydrates.

Write 1 x 4 matrices P, G, and M that represent the nutritional values of each food. Find 3P+2G+2M and tell what the entries represent.

##### Solution Summary

There are several types of problems here. Angular speed, magnitude, trigonometric functions and identities, and matrices and systems of equations are all covered.

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------------- Solution by OTA Dr. Ramas Ramaswami, ID # 102922 ------
Project 3

2. Lance Armstrong won the 2003 Tour de France. The wheel on his bike had a 63 inch diameter. His average speed was 40 km / hour. What was the angular speed of the wheel in revolutions per hour?

1 mile = 5280*12 = 63,360 inches
1 km = 5/8 mile = 5*63,360/8 inches = 39600 inches

Linear velocity = v
Angular velocity = w

v = 40 km/hour = 40*39600 inches/hr
r = 63/2 = 31.5 inches

v = r*w
w = v/r = 40*39600/31.5 = 50286 revolutions/hour
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3. A construction company is making picnic pavilions where the roof will be supported by two sets of beams. The beams in the front are 8 feet high and the back beams are 6.5 feet high. The distance between the front and back beams is 8 feet. At what angle will the roof lay on the front beam?

if the angle is x deg., tan x = 8/(8-6.5)
x = arctan(8/1.5) = 79.4 deg.
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4. Latitude is the north-south location on the earth between the equator and the poles. Since the earth flattens slightly at the poles, a nautical mile varies with latitude. A nautical mile is given by N(?) = 6066 - 31 * cosine 2???? represents the latitude in degrees.
a) What is the length of a British nautical mile at Chicago (latitude of 42 degrees)?
b) What is the length of a British nautical mile at the north pole (latitude of 90 degrees)?
c ) At what latitude north is the length of a British nautical mile found to be 6040 feet?

**** Formula not readable; please send this question in an appropriate format (pdf, MSWord, html...so that I can read the formula) through BrainMass to me and I will solve this problem immediately.

Thanks, Ramas Ramaswami OTA ID# 102922
***************
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5. A guy wire 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.

if the angle is x, tan x = 50/d ==> x = { arctan(50/d) }
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Project 3A
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1. Two cars with new tires are driven at an average speed of 60mph for a test drive of 2000 miles. The diameter of the wheels of one car is 15 inches. The diameter of the wheels of the ...

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