#$&* course MTH 277 09/14/20119:36 am
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Given Solution: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: Find the standard form equation of the sphere with center (-1,2,4) and radius 2. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: (x + 1)^2 + (y - 2)^2 + (z - 4)^2 = 4 confidence rating #$&*:232; 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): Didn’t pay enough attention to the question. Got the initial equation right, but didn’t shift it to standard form ------------------------------------------------ Self-critique rating: 3 ********************************************* 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 X^2 - 2x + y^2 -6y + z^2 +12z = 17 (x - 1)^2 + (y -3)^2 +(z + 6)^2 = 36 + 9 + 17 (x - 1)^2 + (y-3)^2 + (z + 6)^2 = 62 center = (1, 3, -6) radius = sqrt(62) confidence rating #$&*:232; 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Completing the squares we obtain (x^2 - 2 x + 1 - 1) + (y^2 - 6 y + 9 - 9) + (z^2 + 12 z + 36 - 36) = 17 which can be written as (x - 1)^2 - 1 + (y - 3)^2 - 9 + (z + 6)^2 - 36 = 17 and finally as (x - 1)^2 + (y - 3)^2 + (z + 6)^2 = 63 This sphere is centered at (1, 3, -6) and has radius sqrt(63) = 3 sqrt(7). &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Minor arithmetic error when calculating the total from the completed squares ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: Find the standard representation and length of PQ when P = (-3,1,4) and Q = (2,-4,-3). YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Length = sqrt(25 + 25 + 49) Sqrt(99) Length = sqrt(99) Standard representation: (5, -5, 7) confidence rating #$&*:232; 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): Forgot to put the i, j, and k after the appropriate numbers ------------------------------------------------ Self-critique rating: 3 ********************************************* Question: Find a unit vector in the direction of v = <-1, sqrt(3), 4>. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: That is the unit vector confidence rating #$&*:232; 1 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: || v || = sqrt( 1^2 + sqrt(3) ^ 2 + 4^2 ) = sqrt( 26 ) so a unit vector in the direction of v is v / || v ||= < -1, sqrt(3), 4 > / sqrt(26) = <-sqrt(26) / 26, sqrt(78) / 26, 4 sqrt(26) / 26)> . 4 sqrt(26) / 26 is 2 sqrt(26) / 13. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I completely brain-farted on this one. It mistook one term for another. ------------------------------------------------ Self-critique rating:2 ********************************************* Question: Sketch and describe the cylindrical surface given by y = cos x. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: An endless wave. Almost like a sound diagram. confidence rating #$&*:232; 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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): I gave the basic shape, but gave no details, nor direction. Needed more explanation. ------------------------------------------------ Self-critique rating: 2 ********************************************* Question: Determine if u = 2i + 3j + -4k is parallel to v = <1,-3/2,2>. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: no
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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 = sqrt(33) BC = sqrt(13) CA = 6 neither confidence rating #$&*:232; 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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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: #*&!