i have calc 3 online hw its 4 questions and i need help with it

1)Find the length of the curve given by

mathbf{r} left( t right) = frac{sqrt{2}}{2}tmathbf{i} + e^{t/2}mathbf{j} + e^{-t/2}mathbf{k},

where -2 leq t leq 5.


2)Given that the acceleration vector is mathbf{a} left( t right) = left(  -25 cos left( -5 t right) right) mathbf{i} + left( -25 sin  left( -5 t right) right) mathbf{j} + left( -2 t right) mathbf{k}, the initial velocity is mathbf{v} left( 0 right) = mathbf{i + k}, and the initial position vector is mathbf{r} left( 0 right) = mathbf{i +  j + k}, compute:

A. The velocity vector mathbf{v} left( t right) = [removed] mathbf{i} + [removed] mathbf{j} + [removed] mathbf{k}

B. The position vector mathbf{r} left( t right) = [removed] mathbf{i} + [removed] mathbf{j} + [removed] mathbf{k}

Note: the coefficients in your answers must be entered in the form of expressions in the variable emph{t}; e.g. “5 cos(2t)”


3)A gun has a muzzle speed of 50 meters per second. What angle of elevation should be used to hit an object 190 meters away? Neglect air resistance and use g  = 9.8 m / sec^{2} as the acceleration of gravity.


4)If mathbf{r} (t) = -4 t^2 mathbf{i} + 4 t mathbf{j} + (t^2-5 t)mathbf{k} gives the position of a particle at time t, find the time t at which the speed of the particle is minimized.


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