Assignment 8

course mth 164

I did not understand the last question and was unable to draw a diagram.

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008.

Precalculus II

06-30-2007

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16:30:29

Query problem 7.1.B-10 c = 10 , `alpha = 40 deg, right triangle

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RESPONSE -->

ok

confidence assessment:

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16:31:10

what are a, b, and B?

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RESPONSE -->

a is opposite angle A

b is opposit angle b

and B is the third angle in the triangle with angle C being the right angle

confidence assessment: 3

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16:33:35

Explain how you obtained your solutions.

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RESPONSE -->

Given that angle A is 40 degrees then angle B would be 50 degrees since the sum of the angles of a triangle is 180 degrees.

Side a would be 6.43 using sin40=a/10

Side b would be 7.66 using cos40=b/10

confidence assessment: 3

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16:33:49

Query problem 7.1.B-24 cliff height 100 feet, angle of elevation 25 deg. Dist of ship from shore.

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ok

confidence assessment:

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16:34:54

what is the distance so shore?

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RESPONSE -->

Using tan theta = opposite/adjacent

tan(25)=100/x So,

xtan(25)=100

x = 100/tan(25)

x = 214

confidence assessment: 3

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16:35:05

Explain how you obtained your solution

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Given in previous explaination

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16:35:39

describe the triangle you used to find this distance and how you solved it

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I used a right triangle and used the property of tangent being opposite/adjacent sides of the right triangle

confidence assessment: 3

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16:35:58

Query problem 7.1.B-36 guy wire 80 ft long makes an angle of 25 deg with a ground; ht of tower?

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ok

confidence assessment:

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16:36:24

What is the height of the tower?

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RESPONSE -->

27.4 feet

confidence assessment: 3

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16:37:28

Explain how you obtained your solution.

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RESPONSE -->

Setting up a right triangle with the guide wire as the hypotenuse and the angle of elevation 20 degrees as an angle. The trig function sin=opposite/hypotenuse

sin 20 = height of tower/80 feet

solve for height of tower

confidence assessment: 3

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16:38:35

query problem 7.1.A-72 length of ladder around corner hall widths 3 ft and 4 ft `theta relative to wall in 4' hall, ladder in contact with walls

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ok

confidence assessment:

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16:39:17

If the angle is `theta, as indicated, then how long is the ladder?

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RESPONSE -->

If theta is given, then the length of the ladder would be tan (theta) = 3/4

confidence assessment: 2

** In the triangle formed by the ladder in the wider hall, `theta is the angle opposite the 4-foot leg of the triangle. If the length of the part of the ladder in that hall is c1, then c1 = 4 / sin(`theta).

In the triangle formed in the narrower hall, the 3-foot leg of the triangle is parallel to the sides of the wall in the first hall so by corresponding angles `theta is the angle adjacent to that leg, and if c2 is the hypotenuse of that triangle we have c2 = 3 ft / cos(`theta).

The length of the ladder is therefore

3 ft / cos(`theta) + 4 ft / sin(`theta) or

3 ft sec(`theta) + 4 ft csc(`theta). **

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16:39:43

query problem 7.1.A-78 area of isosceles triangle A = a^2 sin`theta cos`theta, a length of equal side

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ok

confidence assessment:

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16:49:41

how can we tell that the area of the triangle is a^2 sin(`theta) cos(`theta)?

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Drawing a peripedicular bisector to one side and dividing the triangle into two right triangles we can find the area of one of the right triangle using A = 1/2 b*h

The base would be 1/2a and the height will be called h.

Using trig ratios and theta a base angle, sin theta = h/a

Solving for h=a sin theta

Using trig ratio for cosine, cos theta = (a/2)/a or cos theta = 1/2

The area of one of these triangle would be A = (1/2)(a/2)(a sin theta) = (1/2)(a^2/2)sin theta

Since there is two of these right triangles making up the isosceles triangle multiply the area by 2 and substitute for 1/2 the cos theta you have

A = (2)(cos theta)(a/2)(a sin theta) = a^2 cos theta sin theta

confidence assessment: 3

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16:50:00

What is the altitude of the triangle?

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RESPONSE -->

altitude is h and is a sin theta

confidence assessment: 3

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16:50:28

If the triangle is divided into two symmetric halves, then what is the length of a base of one of the halves? What therefore is the length of the base of the whole triangle?

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RESPONSE -->

base is a/2

the length of the base of the whole triangle is a

confidence assessment: 3

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16:50:46

What therefore must be the area of the triangle?

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A = a^2sin theta cos theta

confidence assessment: 2

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16:51:07

Query problems 7.2.12 `alpha = 70 deg; `beta = 60 deg, c = 4

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ok

confidence assessment:

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16:53:18

specify the unknown sides and angles of your triangle.

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RESPONSE -->

a = 4.91

b = 4.52

confidence assessment: 3

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16:54:53

Explain how you obtained your results.

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RESPONSE -->

Using the Law of Sines

a/sin70=4/sin50 to obtain a

b/sin60=4/sin50 to obtain b

confidence assessment: 3

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16:55:11

Query problems 7.2.28 b = 4, c = 5, `beta = 40 deg

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k

confidence assessment:

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16:57:15

specify the unknown sides and angles of your triangle.

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RESPONSE -->

angle C = 53.5 degrees

angle A = 86.5 degrees

Side a = 6.21

confidence assessment: 3

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16:58:53

Explain how you obtained your results.

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RESPONSE -->

Using law of sines

sin C/5=sin40/4; solve for sin C and then use the inverse sin key on the calculator to obtain the angle

add angle C and B together and subtract from 180 to find A. Using that value for angle A:

a/sin A = 4/sin 40 then solve for a

confidence assessment: 3

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16:59:20

Query problems 7.2.40 line-of-sight angles 15 deg and 35 deg with line directly to shore

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ok

confidence assessment:

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17:01:28

how far is the ship from each Lighthouse and from the shore?

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RESPONSE -->

I dont understand the question

confidence assessment:

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17:01:40

Explain a you obtained your results.

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RESPONSE -->

I dont understand the question

confidence assessment:

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17:02:11

Comm on any surprises or insights you experienced as a result of this assignment.

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I was fine with the questions except for the last one.

I couldn't draw a diagram to fit the information so I couldn't solve the problem.

confidence assessment:

** First form two right triangles.

The first is from ship to shore to lighthouse A. Angles are 15 deg, 90 deg and 75 deg.

The second is from ship to shore to lighthouse B. Angles are 35 deg, 90 deg and 55 deg.

Now form the triangle from ship to lighthouse A to lighthouse B. Let `alpha be the angle formed at the ship. Then

'alpha = 50deg

'beta = 55deg

'gamma = 75deg

a = 3mi (the separation of the lighthouses).

distance to lighthouse A is the side b:

Law of sines tells us that

sin(50deg)/3 = sin(55deg)/b so

b = 3sin(55deg)/sin(50deg)

b = 3.21 mi.

distance to light house B is side c:

By Law of Sines

c = 3(sin75deg)/sin(50deg)

c = 3.78 mi

distance to shore:

Using first right triangle

Theta = 15

Hypotenuse = distance to light house A = 3.21mi

cos`theta = dist to shore / hypotenuse so

dist to short = hypotenuse * cos(`theta) = 3.21 mi * cos(15 deg) = 3.1 mi.

The same distance would be confirmed by solving the other right triangle. **

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