A space curve is given by for

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

Problem

A space curve is given by

\mathbf r(t)=\langle 3\cos t,\,3\sin t,\,bt\rangle

for

0\le t\le 2\pi

, where

b

is a positive constant measured in meters per radian. (a) Obtain an expression

L(b)

for the exact length of the curve in terms of the parameter

b

. (b) The curve is required to have total length

20\pi

meters over the interval

0\le t\le 2\pi

. Determine the value of

b

that satisfies this condition. (c) Using the value of

b

found in part (b), calculate the particle’s average speed (in meters per second) over the given time interval if time is measured in seconds.