Let . Consider the rose curve given by and the circle given by . Define two…

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

Problem

Let

0 < k < 2

. Consider the rose curve

R

given by

r = 2\sin 2\theta

and the circle

C

given by

r = k

. Define two subregions: • Region

P

is the part of the circle that lies inside the rose. • Region

Q

is the part of the rose that lies outside the circle. The goal is to choose

k

so that the areas of

P

and

Q

are equal. (a) Find the first positive

\theta

, denoted

\theta_0

, at which the curves intersect (

0 < \theta_0 < \pi/2

). (b) Write an exact integral expression for the area of one petal of the rose that lies between

\theta = 0

and

\theta = \pi/2

. (c) Using symmetry, express the total areas of

P

and

Q

in terms of

k

and

\theta_0

, and set up an equation that enforces

\text{Area}(P) = \text{Area}(Q)

. (d) Solve the equation from part (c) for

k

and give the numerical value of

k

accurate to three decimal places.