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question:Sort the following numbers in ascending order: {3, -4, 4, 4 sqrt{2}, 1, frac{29}{pi }}.
answer:The sorted numbers in ascending order are: {-4, 1, 3, 4, 4 sqrt{2}, frac{29}{pi }}.
question:Determine the area of a side-angle-side triangle with sides 12 and 1 and an angle frac{19 pi }{36}.
answer:frac{6 cos left(frac{pi }{36}right)}{sqrt{145+24 sin left(frac{pi }{36}right)}}
question:Expand and find the sum p(x) + q(x) for the following two polynomials: [ p(x) = (3x + 2)^4 ] [ q(x) = (7x + 1)^3 ]
answer:To find the sum, we'll expand both polynomials and then combine like terms: [ p(x) = (3x + 2)^4 = 81x^4 + 216x^3 + 216x^2 + 96x + 16 ] [ q(x) = (7x + 1)^3 = 343x^3 + 147x^2 + 49x + 1 ] Now, adding p(x) and q(x) together: [ p(x) + q(x) = (81x^4 + 343x^3) + (216x^3 + 147x^2) + (216x^2 + 49x) + (96x + 1) + 16 ] Combine like terms: [ p(x) + q(x) = 81x^4 + (216x^3 + 343x^3) + (216x^2 + 147x^2) + (96x + 49x) + (1 + 16) ] [ p(x) + q(x) = 81x^4 + 559x^3 + 363x^2 + 145x + 17 ] So the sum of the polynomials is: [ 81x^4 + 559x^3 + 363x^2 + 145x + 17 ]
question:Determine the angle between two lines with equations y = 0.5x and y = -2x, as measured in the plane defined by these lines.
answer:Take a point A on the first line and a point B on the second line. For instance: A=(1,0.5);quad B=(-1,2). Using the dot product formula, we have: vec{OA}cdotvec{OB}=OAcdot OBcostheta. Rightarrow costheta = frac{vec{OA}cdotvec{OB}}{OAcdot OB} = frac{(1)(-1)+(0.5)(2)}{sqrt{1^2+0.5^2}sqrt{(-1)^2+2^2}} = 0. Therefore, theta = cos^{-1}(0) = 90^circ.