For , let the curves be given by and , with restricted to the first quadrant . The…

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

Problem

For

c > 0

, let the curves be given by

r = 3\sin\theta

and

r = c\cos\theta

, with

\theta

restricted to the first quadrant

0 \le \theta \le \pi/2

. The common region

S(c)

is the set of points that satisfy both inequalities

0 \le r \le 3\sin\theta

and

0 \le r \le c\cos\theta

. Define

A(c)

to be the area of

S(c)

. (a) Show that the curves intersect at exactly one angle

\theta = \theta_c

(0, \pi/2)

and write an equation that

\theta_c

must satisfy in terms of

c

. (b) Express

A(c)

as a single integral in terms of

c

and

\theta_c

, using the smaller of the two radial functions on the appropriate interval. (c) Derive an explicit formula for

A(c)

that depends only on

c

(no

\theta_c

should appear in the final expression). (d) Determine the value of

c

that minimizes

A(c)

. Verify that the critical point you obtain corresponds to a minimum by using either the first- or second-derivative test. (e) Compute the minimum possible area, giving your answer in exact form.

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