Consider the family of outer curves

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

Problem

r_o(\theta)=k\bigl(1+\sin\theta\bigr)

, where

k>1

, together with the fixed inner curve

r_i(\theta)=1+\sin\theta

, both defined for

0\le\theta\le 2\pi

. A region

\mathcal R_k

is enclosed between

r_i

and

r_o

wherever

r_o\ge r_i

. Throughout this problem, do not integrate. (a) Without performing any calculation, explain why

r_o(\theta)\ge r_i(\theta)

for every

\theta

[0,2\pi]

when

k>1

. (b) Describe qualitatively how the distance between the two curves depends on

k

. What happens to that distance if

k

is doubled? (c) Show that doubling

k

multiplies the area of

\mathcal R_k

by a factor strictly greater than

4

. Base your argument on geometric reasoning about radial distances, not on direct area computation. (d) A designer wants the area of

\mathcal R_k

to be exactly nine times the area when

k=1.5

. Explain how to decide whether such a

k

exists and, if so, whether it is unique. Fully justify your answer.