MathDB

Let

n\geq 3

be integer. Given a regular

n

-polygon

P

with side length 4 on the plane

z=0

in the

xyz

-space.Llet

G

be a circumcenter of

P

. When the center of the sphere

B

with radius 1 travels round along the sides of

P

, denote by

K_n

the solid swept by

B

.
Answer the following questions.
(1) Take two adjacent vertices

P_1,\ P_2

P

. Let

Q

be the intersection point between the perpendicular dawn from

G

P_1P_2

, prove that

GQ>1

.
(2) (i) Express the area of cross section

S(t)

in terms of

t,\ n

when

K_n

is cut by the plane

z=t\ (-1\leq t\leq 1)

.
(ii) Express the volume

V(n)

K_n

in terms of

n

.
(3) Denote by

l

the line which passes through

G

and perpendicular to the plane

z=0

. Express the volume

W(n)

of the solid by generated by a rotation of

K_n

around

l

in terms of

n

.
(4) Find

\lim_{n\to\infty} \frac{V(n)}{W(n)} .

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