For , a curve is given by x(t)=k t, y(t)=k t, where is a positive constant to be…

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

Problem

For

0\le t\le \dfrac{\pi}{3}

, a curve is given by

x(t)=k\sin t,\qquad y(t)=k\cos t,

where

k

is a positive constant to be determined. (1) Express the arc length

S(k)

of the curve from

t=0

t=\dfrac{\pi}{3}

in terms of

k

. Do not evaluate numerical constants yet. (2) The arc length is known to be

\dfrac{8\pi}{3}

. Solve for

k

. (3) Using the value of

k

found in part (b), give the exact coordinates of the endpoint of the curve at

t=\dfrac{\pi}{3}