1:Find the volume of the solid in the 1st octant bounded by the coordinate planes,the plane x=3 ,& the parabolic cylinder z= 4-y^2.2 & 3 questions...2: If r with arrow on the top (t)=...

1:Find the volume of the solid in the 1st octant bounded by the coordinate planes,the plane x=3 ,& the parabolic cylinder z= 4-y^2.

2 & 3 questions...

2: If r with arrow on the top (t)= ln(t^2+1)i with ^ on the top +(Tan^-1 t) j with ^ on the top + (t^2 + 1)^1/2 K with ^ on the top is the position of a particle in space at time "t" .Find the angle between the velocity and acceleration vectors at time t=0

3: For the complete revolution( 0≤ t≤ 2π ) of the helix curve: x = Cost, y = Sint and z = t, evaluate . integral with lower limit C y.Sin z ds.

Asked on by sw8-girl

1 Answer | Add Yours

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to find the angle between the velocity vector and the acceleration vector, hence, you need to evaluate the following dot product, such that:

`bar v(t)* bar a(t) = |bar v(t)|*|bar a(t)|*cos (hat(bar v(t),bar a(t)))`

You need to determine the velocity vector, hence, you need to differentiate the position vector, with respect to t, such that:

`(bar r(t))' = ((ln (t^2 + 1)) bar i + (tan^(-1) t) bar j + sqrt(t^2 + 1) bar k)'`

`bar v(t) = (bar r(t))' = (2t)/(t^2+1) bar i + 1/(1 + t^2) bar j + t/(sqrt(t^2+1)) bar k`

You need to determine the acceleration vector, hence, you need to differentiate the velocity vector, with respect to t, such that:

`bar a(t) = ((2t)'(t^2+1) - 2t(t^2+1)')/((t^2+1)^2) bar i - (2t)/((1 + t^2)^2) bar j + (t'sqrt(t^2+1) - (t^2)/(sqrt(t^2+1)))/(t^2+1) bar k`

`bar a(t) = (2t^2 + 2 - 4t^2)/((t^2+1)^2) bar i - (2t)/((1 + t^2)^2) bar j + (sqrt(t^2+1) - (t^2)/(sqrt(t^2+1)))/(t^2+1) bar k`

You need to evaluate `bar v(t)*bar a(t)` such that:

`bar v(t)*bar a(t) = (2t(2 - 2t^2))/((t^2+1)^3) - (2t)/((t^2+1)^3) + t^3/((t^2+1)^3)`

`bar v(t)*bar a(t) = (4t - 4t^3 - 2t + t^3)/((t^2+1)^3)`

`bar v(t)*bar a(t) = (2t - 3t^3)/((t^2+1)^3)`

You need to evaluate `|bar v(t)|` such that:

`|bar v(t)| = sqrt((4t^2)/((t^2+1)^2) + 1/((t^2+1)^2) + t^2/(t^2+1))`

`|bar v(t)| = sqrt((4t^2 + 1 + t^4 + t^2)/((t^2+1)^2))`

`|bar v(t)| = (sqrt(t^4 + 5t^2 + 1))/(t^2+1)`

`|bar a(t)| = sqrt(((2 - 2t^2)^2)/((t^2+1)^4) + (4t^2)/((1 + t^2)^4) + t^4/((t^2+1)^2))`

`|bar a(t)| = (sqrt (4 - 8t^2 + 4t^4 + 4t^2 + t^8 + 2t^6 +t^4))/((1 + t^2)^2)`

`|bar a(t)| = (sqrt (t^8 + 2t^6 + 5t^4 - 4t^2 + 4))/((1 + t^2)^2)`

`cos (hat(bar v(t),bar a(t))) = (|bar v(t)|*|bar a(t)|)/(bar v(t)*bar a(t))`

Hence, evaluating `cos (hat(bar v(t),bar a(t)))` yields `cos (hat(bar v(t),bar a(t))) = (|bar v(t)|*|bar a(t)|)/(bar v(t)*bar a(t)).`

We’ve answered 318,911 questions. We can answer yours, too.

Ask a question