Polar Equation (Archimedean spiral) Help!!?

Question

If i have a polar equation r=0.72 (theta).....and my starting point is (-4,4). How do i find the distance traveled from that starting point to the center?

yeah, i want to find the distance travelled along the curve. I think i'm supposed to use increments of 10 degrees, but i need to find the radius. For this problem, i assumed that if i started at the point (-4,4), it would take 450 degrees to reach the center.

Answer 1

At -4,4, r = 4√2, so θ must be (4√2)/0.72, but this is 5π/2, so (-4,4) cannot lie on the spiral. To have (-4,4) and (0,0) both on the spiral, you need to do both a translation and rotation of coordinate systems. The differential of the length is
ds^2 = dr^2 + (rdθ)^2
ds/dθ = (r^2 + (dr/dθ)^2)^(1/2)
ds/dθ = (0.5184θ^2 + 0.5184)^(1/2)
ds/dθ = 0.72(θ^2 + 1)^(1/2)
ds = 0.72(θ^2 + 1)^(1/2)dθ
s = (1/2)(0.72)[θ(θ^2 + 1)^(1/2) + sinh^-1θ
from θ = 0 to θ = 5π/2
s = (1/2)(0.72)[(5π/2)((5π/2)^2 + 1)^(1/2) + sinh^-15π/2 - (1/2)(0.72)[0(0^2 + 1)^(1/2) - sinh^-10
s = (1/2)(0.72)[(5π/2)((5π/2)^2 + 1)^(1/2) + sinh^-15π/2
s = 23.37884

Edit:
450° is correct, but you need to work in radians

Answer 2

There is some confusion here. The point (-4,4) is not on the curve of the function r = 0.72(theta) as theta = -4 gives
r = -2.88 assuming that (theta,r) are the polar cordinates. Are the (-4,4) cartesian coordinates? Do you want to travel along the curve? If so this could be a tricky integral. Do you need to travel from (-4,4) to the nearest point on the curve first?