Consider the family of solids indexed by the positive integer

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Consider the family of solids

\{V_n\}

indexed by the positive integer

n

. Each solid

V_n

has its base region in the

xy

-plane equal to the rectangle

0\le x\le L

0\le y\le W

, where

L

and

W

are positive real constants. Beginning at

x=0

and moving toward

x=L

, cross sections taken perpendicular to the

x

-axis repeat a two-segment cycle of total length

d

units: 1. On the first sub-interval of length

\dfrac{d}{2}

the cross section is a square whose side length equals

W

. 2. On the next sub-interval of length

\dfrac{d}{2}

the cross section is a rectangle whose height is

kW

for some fixed constant

k>0

and whose base equals

W

. Exactly

n

complete cycles fit in the interval

0\le x\le L

, so that

L=nd

. (a) Derive a closed-form expression for the volume

V_n

in terms of

n, d, W,

and

k

. (b) Express

V_n

solely as a function of

L, W,

and

k

by eliminating

d

and

n

. (c) Suppose

k=\tfrac{1}{3}

W=2

, and

L=15

. Compute the exact numerical value of the corresponding volume. (d) Holding

L

and

W

fixed, determine the value of

k

that minimizes the volume and justify that your result gives a global minimum on

k>0