Final answer:
To sketch the curve 8*Cos(t)i+8*Sin(nt)j+t, one should plot the cos and sin components for x and y. To find the arclength over [0, π/2], calculate the derivatives of these components, square them, add them, and integrate the square root of this sum over the interval.
Step-by-step explanation:
The question requires us to sketch the curve represented by the vector function 8*Cos(t)i+8*Sin(nt)j+t, and find its arclength over the interval [0, π/2]. The curve appears to be a parametric equation in two dimensions (i and j components) with a possible helical or sinusoidal shape in the xy-plane. To sketch the curve, one would need to plot points derived from the functions 8*Cos(t) and 8*Sin(nt) in the respective x and y coordinates, and trace the path as t varies from 0 to π/2.
To calculate the arclength of the curve, we use the formula for arclength in parametric equations:
S = ∫ √[(dx/dt)^{2} + (dy/dt)^{2}] dt
where x(t) = 8*Cos(t) and y(t) = 8*Sin(nt). We compute the derivatives dx/dt and dy/dt, then integrate the square root of the sum of their squares over the given interval.