answer:After calculating, the polygon's specifications are as follows: - Area: 0.27 square units - Interior Angles: {2.76, 1.68, 1.09, 3.02, 2.93, 1.08} radians - Perimeter: 2.3 units Based on these calculations, the polygon is classified as 'Convex'.

answer:The addition of matrices ( A ) and ( B ) is performed element-wise. Here is the calculation for each element: 1. For the first row: [ begin{align*} C_{11} &= A_{11} + B_{11} = -frac{3}{8} + frac{27}{8} = 3 C_{12} &= A_{12} + B_{12} = -frac{59}{8} - frac{47}{8} = -frac{53}{4} end{align*} ] 2. For the second row: [ begin{align*} C_{21} &= A_{21} + B_{21} = 9 - frac{49}{8} = frac{72}{8} - frac{49}{8} = frac{23}{8} C_{22} &= A_{22} + B_{22} = -frac{7}{8} + frac{17}{2} = -frac{7}{8} + frac{68}{8} = frac{61}{8} end{align*} ] 3. For the third row: [ begin{align*} C_{31} &= A_{31} + B_{31} = frac{33}{4} + frac{49}{8} = frac{66}{8} + frac{49}{8} = frac{115}{8} C_{32} &= A_{32} + B_{32} = frac{25}{8} + frac{5}{4} = frac{25}{8} + frac{10}{8} = frac{35}{8} end{align*} ] 4. For the fourth row: [ begin{align*} C_{41} &= A_{41} + B_{41} = -frac{31}{8} - frac{19}{2} = -frac{31}{8} - frac{76}{8} = -frac{107}{8} C_{42} &= A_{42} + B_{42} = -frac{17}{4} - frac{43}{8} = -frac{34}{8} - frac{43}{8} = -frac{77}{8} end{align*} ] Therefore, the matrix ( C ) is: [ C = left( begin{array}{cc} 3 & -frac{53}{4} frac{23}{8} & frac{61}{8} frac{115}{8} & frac{35}{8} -frac{107}{8} & -frac{77}{8} end{array} right) ]

answer:To find the volume, we'll use cylindrical coordinates ({eq}r, theta, z {/eq}). The boundaries are defined by: {eq}begin{align*} z &= -2x^2 - 2y^2 + 10 quad text{(paraboloid)} z &= 4 quad text{(plane)} end{align*} {/eq} From the paraboloid equation, we get: {eq}-2(x^2 + y^2) + 10 = 4 x^2 + y^2 = 3 r^2 = 3 r = sqrt{3} quad text{(for} r ge 0 text{)} {/eq} The volume integral is set up as: {eq}begin{align*} V &= int_0^{2pi} int_0^{sqrt{3}} int_4^{-2r^2 + 10} r , dz , dr , dtheta V &= int_0^{2pi} int_0^{sqrt{3}} [rz]_4^{-2r^2 + 10} , dr , dtheta V &= int_0^{2pi} int_0^{sqrt{3}} r(-2r^2 + 10 - 4) , dr , dtheta V &= int_0^{2pi} int_0^{sqrt{3}} (-2r^4 + 6r^2) , dr , dtheta V &= int_0^{2pi} left[frac{-2r^5}{5} + frac{6r^3}{3}right]_0^{sqrt{3}} , dtheta V &= int_0^{2pi} left(frac{-2cdot 3^5}{5} + frac{6cdot 3^3}{3}right) , dtheta V &= int_0^{2pi} left(frac{-54}{5} + 54right) , dtheta V &= int_0^{2pi} frac{216}{5} , dtheta V &= left[frac{216theta}{5}right]_0^{2pi} V &= frac{216}{5} cdot 2pi V &= frac{432pi}{5} end{align*} {/eq} Therefore, the volume of the solid is {eq}frac{432pi}{5} {/eq} cubic units.

answer:To determine if the system has a unique solution, a dependent solution, or is inconsistent, calculate the determinant of the coefficient matrix. If the determinant is non-zero, the system has a unique solution. If the determinant is zero, the system can either have infinitely many solutions or be inconsistent, depending on the values of the variables in the augmented part. The determinant of the coefficient matrix before row reduction is: [ text{Det}(A) = left| begin{array} {ccc} 1 & 2 & 2 1 & a & 3 1 & 11 & a end{array} right| ] The determinant is found using cofactor expansion along the first row: [ text{Det}(A) = 1left| begin{array} {cc} a & 3 11 & a end{array} right| - 2left| begin{array} {cc} 1 & 3 1 & a end{array} right| + 2left| begin{array} {cc} 1 & a 1 & 11 end{array} right| ] Simplify the determinant: [ text{Det}(A) = (a^2 - 33) - (2a - 6) + 2(a - 11) ] [ text{Det}(A) = a^2 - 4a - 5 ] Set the determinant equal to zero and solve for a: [ (a - 5)(a + 1) = 0 ] So a = 5 or a = -1. For these values, the system either has no solution or infinitely many solutions. To determine the specific behavior for each value: 1. If a = 5, the second and third rows become identical, which makes the system inconsistent, as we would have a row of zeros in the reduced matrix. 2. If a = -1, the second row becomes the negative of the first row, which indicates dependency. In this case, the system would have infinitely many solutions. For all other values of a, the system has a unique solution, as the determinant is non-zero.