Let and be fixed real numbers

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Let

a>0

and

b\ne 0

be fixed real numbers. A helix is given by

\mathbf r(t)=\langle a\cos t,\,a\sin t,\,bt\rangle

. (a) Show that the speed

|\mathbf r'(t)|

is constant and determine its value in terms of

a

and

b

. (b) Derive a formula for the curvature

\kappa

of the helix expressed solely in terms of

a

and

b

. (c) State whether the curvature depends on

t

. Explain the geometric significance of your conclusion regarding the shape of the helix. (d) For

a=5

and

b=2

, compute the radius of curvature of the helix.