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Math141Sols4S11

Math141Sols4S11 - MIDTERM 4 SOLUTIONS(CHAPTERS 5 AND 6 MATH...

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Unformatted text preview: MIDTERM 4 SOLUTIONS (CHAPTERS 5 AND 6) MATH 141 – SPRING 2011 – KUNIYUKI 150 POINTS TOTAL: 57 FOR PART 1, AND 93 FOR PART 2 Show all work, simplify as appropriate, and use “good form and procedure” (as in class). Box in your final answers! No notes or books allowed. • Write units in your final answers where appropriate. • Try to avoid rounding intermediate results; if you do round off, do it to at least five significant digits. • We assume that all vectors on this test are in the usual xy-plane. PART 1: SCIENTIFIC CALCULATORS ALLOWED! (57 POINTS TOTAL) 1) Find the length of Side c for the triangle below using the Law of Sines. Round off your answer to the nearest tenth (i.e., to one decimal place) of an inch. (9 points) Find Angle B . B = 180 ¡ ¡ 80 ¡ ¡ 44 ¡ = 56 ¡ . Use the Law of Sines. We now know B , b , and C , and we want to find c . b sin B = c sin C 41 sin56 ¡ = c sin44 ¡ c = 41sin44 ¡ sin56 ¡ c ¡ 34.4 inches This makes sense, because c is shorter than b , and Angle C is smaller than Angle B . Remember that smaller angles face (“eat”) shorter sides in a triangle. 2) A slanted lightning rod, represented by line segment BC in the figure below, has length 17.1 feet. An observer stands at point A . The observer’s shoes are 12.7 feet from the base of the rod and are 23.4 feet from the top of the rod. Find the measure of Angle A , the angle of elevation from the observer’s shoes to the top of the rod, using the Law of Cosines. Round off your answer to the nearest tenth of a degree. Note: Angle C is obtuse, not right. (10 points) Use the Law of Cosines. a 2 = b 2 + c 2 ¡ 2 bc cos A ¢ cos A = b 2 + c 2 ¡ a 2 2 bc cos A = 12.7 ( ) 2 + 23.4 ( ) 2 ¡ 17.1 ( ) 2 2 12.7 ( ) 23.4 ( ) = 416.44 594.36 ¢ 0.70065 £ ¤ ¥ ¦ § ¨ © A = cos ¡ 1 416.44 594.36 £ ¤ ¥ ¦ § ¨ (in degrees) A ¢ 45.5 ¡ (Make sure you are in degree mode when you press the cos ¡ 1 button. You don’t have to worry about “Quadrant issues” here, due to the range of the inverse cosine function..) Note: B ¡ 32.0 ¡ and C ¡ 102.5 ¡ . 3) Consider the vectors v and w , where v = 7, 2 and w = ¡ 3,12 . (9 points total) a) Find the vector v • w ( ) w . Write your answer in x , y component form. (5 points) Observe that v • w is a scalar, and v • w ( ) w , which is a scalar times a vector, is a vector. v • w = 7, 2 • ¡ 3,12 = 7 ( ) ¡ 3 ( ) + 2 ( ) 12 ( ) = ¡ 21 + 24 = 3 Therefore, v • w ( ) w = 3 w = 3 ¡ 3,12 = 3 ¡ 3 ( ) , 3 12 ( ) = ¡ 9, 36 b) Does v • w ( ) w have the same direction as the vector w ? (2 points) Y e s N o Since w is a nonzero vector, then any positive scalar times w is a vector that has the same direction. the same direction....
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Math141Sols4S11 - MIDTERM 4 SOLUTIONS(CHAPTERS 5 AND 6 MATH...

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