#$&* course Mth 164 OLUTIONS/COMMENTARY ON QUERY 8
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10:55:51 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: we want to find the other sides of the triangle so use c as the hypotenuse hypotenuse * sin(theta) = a a = 6.4 b = c * cos(alpha) = 7.7 it is a right triangle so it should be 90. so to get the other angle do 90 - 40 which leaves you with 50 deg. confidence rating #$&* 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** c is the hypotenuse. The angle a opposite the angle alpha satisfies hypotenuse * sin(alpha) = a so that a = 10 * sin(40 deg) = 6.43, approx.. We also have b = c * cos(alpha) = 10 * cos(40 deg) = 7.66 approx.. The remaining angle of the triangle is beta = 90 deg - alpha = 90 deg - 40 deg = 50 deg. **
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*Ok **** Query problem 7.1.B-24 cliff height 100 feet, angle of elevation 25 deg. Dist of ship from shore.
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YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: we want to find the adjacent side of the triangle being made so: adjacent = opposite/ tan(theta) = 100ft/tan(25) = 215 ft. confidence rating #$&*3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** The cliff height forms a leg of a right triangle, oppposite the 25 deg angle. The distance from ship to shore forms the other leg of the triangle, adjacent to the 25 deg angle. Cliff height / distance from ship to shore = opposite side / adjacent side = tan(25 deg) so adjacent side = opposite side / tan(25 deg) = 100 ft / tan(25 deg) = 214.5 ft. **
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*Ok **** Query problem 7.1.B-36 guy wire 80 ft long makes an angle of 25 deg with a ground; ht of tower?
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11:07:20 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: we want the height of the tower so to do this we will take the hypotenuse and use the sin of the angle to find the other side of the triangle. hypotenuse * sin(theta) = 80ft * sin(25) = 34 ft confidence rating #$&*3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** The guy wire is the hypotenuse of a right triangle for which the altitude is opposite the 25 degree angle. Thus we have altitude = hypotenuse * sin(25 deg) = 80 ft * sin(25 deg) = 33.8 ft. **
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*Ok **** query problem 7.1.A-72 length of ladder around corner hall widths 3 ft and 4 ft `theta relative to wall in 4' hall, ladder in contact with walls
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YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: here we are looking for the hypotenuse so we must use the a and b sides of the wall to find c. So: 3ft/cos(theta) + 4ft/sin(theta) will give us the other length. confidence rating #$&*2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** 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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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*Ok **** query problem 7.1.A-78 area of isosceles triangle A = a^2 sin`theta cos`theta, a length of equal side
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YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: A = 1/2 * base * height = 1/2 a^2sin(theta) cos(theta) = 2(1/2a^2sin(theta)cos(theta) = a^2sin(theta)cos(theta) confidence rating #$&*2.5 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** If you divide the triangle by its axis of symmetry you get two congruent right triangles, each with angle `theta opposite the altitude and adjacent to the base. The side a makes up the hypotenuse of either of these triangles. The altitude of each is therefore a sin(`theta) and the base is a cos(`theta). The area of each triangle is thus 1/2 * base * height = 1/2 a sin(`theta) a cos(`theta) = 1/2 a^2 sin(`theta) cos(`theta). The areas of the two right triangles add up to the area of the isosceles triangle. This area is therefore 2 ( 1/2 a^2 sin(`theta) cos(`theta) ) = a^2 sin(`theta) cos(`theta). ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*Ok **** Query problems 7.2.12 alpha = 70 deg; `beta = 60 deg, c = 4
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11:48:52 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: we can easily find the third angle my taking the two we already have and taking them from 180 deg. If we do so we are left with 50 degs for the other angle. we can find the sides now that we have all of the angle using sin and cos. first for b: sin(beta)/b = sin(gamma)/c sin(60)/b = sin(0)/4 b= 4.5 a: sin(alpha)/a = sin(gamma)/c sin(70)/a = sin(50)/c a = 4.9 confidence rating #$&*3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** GOOD STUDENT SOLUTION: if alpha = 70 deg; and `beta = 60 deg, then `gamma = 50 deg alpha + `beta + `gamma = 180 deg 70 deg + 60 deg = `gamma = 180 deg. `gamma = 180 deg - 130 deg `gamma = 50 deg. Now for the sides - knowing what the three angles are and knowing that c = 4, a = : sin alpha / a = sin`gamma / c sin70 deg / a = sin 50 deg / 4 a = 4(sin70 deg) / sin50 deg a is approx. 4.91 b = : sin `beta / b = sin `gamma/ c sin 60 deg / b = sin 50 deg / 4 b = 4(sin 60 deg) / sin 50 deg b is approx. 4.52 **
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*Ok **** Query problems 7.2.28 b = 4, c = 5, `beta = 40 deg
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YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: we can simply find that the sin of gamma will be about .8 and using this we can say that the arcsin(.8) will be about 53 deg. gamma = 53 deg so that means alpha will equal 87 deg. a = sin(87) * 4/sin(40) = 6.2 confidence rating #$&*2.5 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** sin(`gamma) = .80. Thus `gamma = arcsin(.80) = 53 deg, approx., or 180 deg - 53 deg = 117 deg. Note that we have to consider both angles because the sine doesn't distinguish between the first and second quadrant, whereas the cosine (which is negative in the second quadrant) would. If `gamma = 53 deg then alpha would be 87 deg. In this case the Law of Sines tells us that a = sin(87 deg) * 4 / sin(40 deg) = 6.2, approx.. If `gamma = 117 deg then alpha would be 23 deg so that a = sin(23 deg) * 4 / sin(40 deg) = 2.8 or so. You should draw both triangles to see that both of these solutions are possible. **
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I did not think about there being a second possibility to this. ------------------------------------------------ Self-critique rating #$&*Ok **** specify the unknown sides and angles of your triangle.
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11:57:48 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I do not understand this question. The unknown sides to what triangle? confidence rating #$&*0 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: if it is possible to draw the triangle or even if it isn't we can solve for a. so we can say alpha+beta + gamma=180 deg. so alpha + 40+.80=180 so alpha= 139.2 we can then find the value of a by saying sin 139.2/a= sin 40/4 which is 4 sin 139.2= a sin 40 deg so a= 4.07 and alpha=139.2
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11:57:48
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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&* **** Query problems 7.2.40 line-of-sight angles 15 deg and 35 deg with line directly to shore points are 3 miles apart .
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YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: alpha = 50 deg beta = 55 deg gamma = 75 deg side a = 3 mi side b: sin(50)/3 = sin(55)/b b =3.2 mi side c: 3(sin(75))/sin(50)= 3.8 confidence rating #$&*2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** 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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&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I wasn’t for sure about using two different triangles but I figured some of it out but I do not fully understand that last part of the given answer.