#$&* course Mth 277 9/9/12 11pm. If your solution to stated problem does not match the given solution, you should self-critique per instructions athttp://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: Find the standard form equation of the sphere with center (-1,2,4) and radius 2. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2. (x+1)^2 + (y-2)^2 + (z-4)^2 = 4. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: A point (x, y, z) is on the given sphere if its distance from (-1, 2, 4) is 2, so that sqrt( (x - (-1))^2 + (y - 2)^2 + (z - 4)^2 ) = 2 and (x + 1)^2 + (y - 2)^2 + (z - 4)^2 = 4. This is the equation of the sphere in one form. Expanding the squares we obtain x^2 + 2 x + 1 + y^2 - 4 y + 4 + z^2 - 8 x + 16 = 4 which we rearrange to the standard form x^2 + 2 x + y^2 - 4 y + z^2 - 8 z + 13 = 0. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: Find the center and radius of the sphere with equation x^2 + y^2 + z^2 - 2x - 6y + 12z - 17 = 0. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x^2 + y^2 + z^2 - 2x - 6y + 12z - 17 = 0. After completing the square. (x-1)^2 + (y-3)^2 + (z+6)^2 = 63. This is the standard form for a sphere. Center = (1, 3, -6). Radius = sqrt(63) = 7.94. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Completing the squares we obtain (x^2 - 2 x + 1 - 1) + (y^2 - 6 y + 9 - 9) + (z^2 + 12 z + 36 - 36) = 17 which can be written as (x - 1)^2 - 1 + (y - 3)^2 - 9 + (z + 6)^2 - 36 = 17 and finally as (x - 1)^2 + (y - 3)^2 + (z + 6)^2 = 63 This sphere is centered at (1, 3, -6) and has radius sqrt(63) = 3 sqrt(7). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating:OK ********************************************* Question: Find the standard representation and length of PQ when P = (-3,1,4) and Q = (2,-4,-3). YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: sqrt( (2+3)^2 + (-4-1)^2 + (-3-4)^2 ). sqrt( 25 + 25 + 49) 3* sqrt(11) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: PQ = (2 - (-3) ) i + (-4 - 1) j + (-3 - 4) k = 5 i - 5 j - 7 k. || PQ || = sqrt( 5^2 + 5^2 + 7^2) = sqrt(99). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK ********************************************* Question: Find a unit vector in the direction of v = <-1, sqrt(3), 4>. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (-i + sqrt(3)j + 4k) / sqrt(1^2 + sqrt(3)^2 + 4^2). (-i + sqrt(3)j + 4k)/ (2* sqrt(5) ) confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: || v || = sqrt( 1^2 + sqrt(3) ^ 2 + 4^2 ) = sqrt( 26 ) so a unit vector in the direction of v is v / || v ||= < -1, sqrt(3), 4 > / sqrt(26) = <-sqrt(26) / 26, sqrt(78) / 26, 4 sqrt(26) / 26)> . 4 sqrt(26) / 26 is 2 sqrt(26) / 13. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK but sqrt( 1^2 + sqrt(3) ^ 2 + 4^2 ) does not equal sqrt(26). ------------------------------------------------ Self-critique rating: OK ********************************************* Question: Sketch and describe the cylindrical surface given by y = cos x. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: y= cosx y= cos x is a sinusoidal wave starting at y=1, x=0. So I suppose the surface would just be this wave projected downard along the z axis. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: In the x-y plane y = cos(x) consists of a sinusoidal function oscillating between the lines y = -1 and y = 1, with period 2 pi radians, and containing the point (0, 1). The surface in 3 dimensions repeats this same curve for every value of z, so that the graph represents a wavy curtain hanging vertically downward, intersecting the xy plane along the sinusoidal curve. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: Determine if u = 2i + 3j + -4k is parallel to v = <1,-3/2,2>. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: If you divided u by 2 you would get i + 3/2j - 2k. The signs don't match up with v, therefore v and u aren't parallel. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Two vectors are parallel if the angle between them is 0 or pi radians (180 degrees), meaning that the cosine of the angle between them is 1 or -1. u dot v = || u || || v || cos(theta) so that cos(theta) = u dot v / (|| u || || v || ) = (2 * 1 + 3 * (-3/2) + (-4 * 2) ) / ( sqrt(2^2 + 3^2 + 4^2) * sqrt( 1^2 + (3/2)^2 + 2^2) ) = (-21/2) / (sqrt( 29) sqrt(29/4). This is not 1 or -1, so the cosine is neither 0 nor pi rad (i.e., 180 deg). The vectors are therefore not parallel. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating: OK ********************************************* Question: Find the lengths of the sides of the triangle and determine if the triangle with vertices A(3,0,0), B(7,1,4) and C(5,4,4) is a right triangle, isosceles triangle, both, or neither. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: AB = 4i + j + 4k. length = sqrt(33) BC = -2i +3j + 0k. length = sqrt(13) AC = 2i + 4j + 4k. length = 6. No equal lengths so not isosceles. If you take arcsin (sqrt(13)/6) you don't get a 30 or 60 degree angle. confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: The sides can be represented by the vectors AB = < 4, 1, 4 >, BC = < -2, 3, 0 > and AC = < 2, 4, 4 >. The magnitudes of these vectors are respectively sqrt(33) sqrt(13) sqrt(36). None of the sides are the same length so the triangle is not isosceles. The sum of the squares of the shorter two side is 33 + 13 = 46, which is not equal to the sum of the longest, so the triangle is not a right triangle. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique rating:OK " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!