A spiral is described in polar coordinates by r( )= k , 0 2

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

Problem

A spiral is described in polar coordinates by

r(\theta)= k\theta,\quad 0 \le \theta \le 2\pi,

where

k>0

is a constant. Designers need the length of one complete turn of the spiral (from

\theta=0

\theta=2\pi

) to be exactly

30\,\text{cm}

. (a) Derive a formula for the length

L(k)

of the spiral’s first turn in terms of

k

. (b) Solve explicitly for

k

so that

L(k)=30

. Express your answer exactly, then provide a decimal approximation to three significant figures. (c) Suppose the value of

k

you found is increased by 5%. Use differentials or another appropriate method to estimate the resulting percentage change in the spiral’s length. Interpret your result.

Related theory

Polar Functions
Parametric Functions

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