PROFESSOR: Find the arc length function for the helix r of t equals angle bracket 3 cosine t comma 3 sine t comma 4t, close angle bracket, the restriction that t is greater than or equal to 0. Then use the relationship between arc length and the parameter t to find an arc length parameterization of r of t.

So the arc lung function we're going to be using is s of t equals the integral from a to t of the magnitude of r prime of u du. So let's find r prime of u first. So r prime of u would be equal to angle bracket minus 3 sine u comma 3 cosine u comma 4.

And then the magnitude of r prime of u be equal to the square root of 9 sine squared t plus 9 cosine squared t plus 16. Now, knowing that sine squared t plus cosine squared is 1, this becomes the square root of 25, which is equal to 5.

So s of t, going back to our arc length formula, is the integral from 0 to t of 5 du, which is going to be equal to 5t. So there's our arc length function for a given value of t.

Now, because we know that arc length s is equal to 5t, this means that t is equal to s over 5, which means we can rewrite r of t, vector valued function r of t, equal to angle bracket 3 cosine of s over 5, comma 3 sine of s over 5, comma 4s over 5. Close angle bracket.