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question:List the vectors that form a basis for the null space of the following matrix, rounding your answer to three decimal places: left( begin{array}{ccccc} 3 & -3 & 0 & -8 & -2 end{array} right)
answer:The null space basis vectors are: begin{align*} v_1 &= begin{pmatrix} 0 0 1 0 0 end{pmatrix} v_2 &= begin{pmatrix} 1 1 0 0 0 end{pmatrix} v_3 &= begin{pmatrix} 2 0 0 0 3 end{pmatrix} v_4 &= begin{pmatrix} 8 0 0 3 0 end{pmatrix} end{align*}
question:Find all real solutions to the equation frac{-3x^2 + frac{71x}{3} - frac{2}{3}}{-frac{41x^2}{3} + frac{17x}{3} - frac{71}{3}} = 0.
answer:To find the real solutions to the given equation, we need to set the numerator equal to zero, as the denominator cannot be zero to avoid division by zero. [ -3x^2 + frac{71x}{3} - frac{2}{3} = 0 ] To solve this quadratic equation, we can multiply through by 3 to clear the fractions: [ -9x^2 + 71x - 2 = 0 ] Next, we can apply the quadratic formula: [ x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ] where a = -9, b = 71, and c = -2. Plugging in the values, we get: [ x = frac{-71 pm sqrt{71^2 - 4(-9)(-2)}}{2(-9)} ] [ x = frac{-71 pm sqrt{4969 - 72}}{-18} ] [ x = frac{-71 pm sqrt{4897}}{-18} ] Simplifying the square root: [ x = frac{-71 pm sqrt{49 cdot 101}}{-18} ] [ x = frac{-71 pm 7sqrt{101}}{-18} ] So, the real solutions are: [ x = frac{-71 - 7sqrt{101}}{-18}, x = frac{-71 + 7sqrt{101}}{-18} ] Rationalizing the denominators, we get: [ x = frac{71 + 7sqrt{101}}{18}, x = frac{71 - 7sqrt{101}}{18} ] The solutions are: [ left{left{xto frac{71 + 7sqrt{101}}{18}right},left{xto frac{71 - 7sqrt{101}}{18}right}right} ]
question:In 2013, the Barton and Barton Company transitioned from using the FIFO (First-In, First-Out) method to the average cost method for inventory valuation. As of December 31, 2012, B & B's FIFO-based inventory value was 32 million. However, under the average cost method, the inventory would have been valued at 23.8 million. What journal entry should B & B make in 2013 to reflect this adjustment? Additionally, outline the other necessary steps for reporting this change in accounting method.
answer:The adjustment in 2013 is due to the overstatement of inventories by 8.2 million (32 million - 23.8 million) under the FIFO method compared to the average cost method, leading to inflated profits by the same amount. To rectify this error, B & B should record the following journal entry: Retained Earnings | Dr. | 8.2 million Inventories | Cr. | 8.2 million This entry corrects the retained earnings, as the overstated profits from the previous year have been included in it. In addition to this entry, B & B must restate its 2012 financial statements using the average cost method for inventory valuation. This ensures consistency and comparability in the financial reporting. The company should provide a disclosure note explaining the change in accounting policy and the impact on the financial statements for both the current and prior years.
question:A triangle has one angle measuring 83 {}^{circ}, a side of length 4, and another angle measuring 3 {}^{circ}. Determine the centroid, circumcenter, incenter, orthocenter, nine-point center, and symmedian point of the triangle.
answer:Centroid: left(frac{4}{3} (sin (3 {}^{circ}) sin (7 {}^{circ}) sec (4 {}^{circ})+1), frac{4}{3} sin (3 {}^{circ}) cos (7 {}^{circ}) sec (4 {}^{circ})right) Orthocenter: (4 sin (3 {}^{circ}) sin (7 {}^{circ}) sec (4 {}^{circ}), 4 sin (7 {}^{circ}) cos ({}^{circ}) (2 cos (2 {}^{circ})-1) sec (4 {}^{circ})) Nine-Point Center: (2 sin (3 {}^{circ}) sin (7 {}^{circ}) sec (4 {}^{circ})+1, 2 (sin ({}^{circ})-sin (3 {}^{circ})+sin (5 {}^{circ})-sin (7 {}^{circ})+sin (9 {}^{circ})) cos ({}^{circ}) sec (4 {}^{circ})) Circumcenter: (2, -2 tan (4 {}^{circ})) Incenter: left(frac{4 sin left(frac{3 {}^{circ}}{2}right) left(sin left(frac{7 {}^{circ}}{2}right)+cos left(frac{7 {}^{circ}}{2}right)right)}{cos (2 {}^{circ})-sin (2 {}^{circ})}, frac{4 sin (3 {}^{circ}) cos (7 {}^{circ}) sec (4 {}^{circ})}{cos (7 {}^{circ}) sec (4 {}^{circ})+sin (3 {}^{circ}) sec (4 {}^{circ})+1}right) Symmedian Point: left(frac{8 sin ^2({}^{circ}) (9 cos (2 {}^{circ})+6 cos (4 {}^{circ})+3 cos (6 {}^{circ})+3 cos (8 {}^{circ})+2 cos (10 {}^{circ})+cos (12 {}^{circ})+6)}{-cos (6 {}^{circ})+cos (8 {}^{circ})+cos (14 {}^{circ})+3}, frac{8 sin (3 {}^{circ}) cos (4 {}^{circ}) cos (7 {}^{circ})}{-cos (6 {}^{circ})+cos (8 {}^{circ})+cos (14 {}^{circ})+3}right)