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question:In a side-angle-side triangle, the lengths of two sides are 13 units and 10 units, and the included angle measures 146 {}^{circ}. Find the coordinates of the triangle's vertices and the measures of the three interior angles.
answer:The vertices of the triangle can be expressed as: A = {0,0} B = left{frac{130 cos (34 {}^{circ})+100}{sqrt{260 cos (34 {}^{circ})+269}}, frac{130 sin (34 {}^{circ})}{sqrt{260 cos (34 {}^{circ})+269}}right} C = left{sqrt{260 cos (34 {}^{circ})+269}, 0right} The measures of the interior angles (in radians) can be computed as: angle A = cos ^{-1}left(frac{13 cos (34 {}^{circ})+10}{sqrt{260 cos (34 {}^{circ})+269}}right) angle B = cos ^{-1}left(frac{10 cos (34 {}^{circ})+13}{sqrt{260 cos (34 {}^{circ})+269}}right) angle C = cos ^{-1}left(frac{-130 (sin (22 {}^{circ})+1)-269 cos (34 {}^{circ})}{sqrt{33800 sin (22 {}^{circ})+139880 cos (34 {}^{circ})+106161}}right) Please note that the angles must be converted to degrees if required for a specific context.
question:Which of the following statements best describes the role of a catalyst in a heterogeneous catalytic reaction? A) The catalyst lowers the free energy of activation, making the reaction more favorable, and remains unchanged at the end of the reaction. B) A small amount of catalyst can significantly convert a large amount of reactants, but the concentration of the catalyst does not affect the reaction rate. C) The catalyst does not form an intermediate complex with the reactants, thus not altering the reaction mechanism. D) The surface of the catalyst is insignificant during the reaction and does not contribute to the catalytic activity.
answer:A) The catalyst lowers the free energy of activation, making the reaction more favorable, and remains unchanged at the end of the reaction.
question:Determine the equation of the surface that contains all points whose coordinates satisfy the relationships derived from the parametric equations of the given surface: {eq}rleft( {u,v} right) = leftlangle {4u, - 2v,{u^2} + v,6v - u} rightrangle {/eq}
answer:From the given parametric equations, we have: {eq}displaystyle { v = frac{{x + 4z}}{{22}} u = frac{{3x + z}}{{11}} } {/eq} Substituting these into the equation {eq}y = {u^2} + v {/eq}, we obtain the equation of the surface: {eq}displaystyle y = {left( {frac{{3x + z}}{{11}}} right)^2} + frac{{x + 4z}}{{22}} {/eq}
question:What is the meaning of radical form in mathematics?
answer:Radical form in mathematics refers to the representation of a number or expression using radicals, which are square roots or higher-order roots. It involves simplifying the expression so that there are no fractional exponents or nested radicals. The goal is to express the number or expression in its simplest and most concise radical form.