Final answer:
To find the total mass and moments of inertia of the solid with the given mass density occupying the unit cube in the first octant, you need to integrate over the cube and use the formulas for mass and moments of inertia. The mass density is given by xk + 5 kg/m, and you can substitute the values into the respective formulas to calculate the total mass and moments of inertia.
Step-by-step explanation:
The total mass M and the moments of inertia Ix, Iy, and Iz of the solid can be found by integrating over the unit cube and using the given mass density. The mass is given by M = ∫∫∫ ρ(x, y, z) dV, where ρ(x, y, z) = xk + 5 kg/m. Substituting in the values, we get M = ∫[0, 1]∫[0, 1]∫[0, 2] (xk + 5) dxdydz. To find the moments of inertia, we use the formulas Ix' = ∫∫∫ ρ(x, y, z)(y2 + z2) dV, Iy'' = ∫∫∫ ρ(x, y, z)(x2 + z2) dV, and Iz = ∫∫∫ ρ(x, y, z)(x2 + y2) dV. Substituting the given values, we can simplify these expressions and evaluate the integrals to find the moments of inertia.