Calculus: Analytic Geometry and Calculus, with Vectors

5.2 Trends, maxima, and minima 299

Our first step is to draw Figure 5.291 and look at it.^24
Our good sense should tell us that if the squares are -----------------
small, then the tank will be shallow and the volume will
be small. Taking larger squares should yield greater 1;^115
volumes unless we make the squares so large that the
area of the base of the tank is small enough to overcome
the advantage of making the tank deeper. To become Figure 5.
quantitative about this matter, we let x denote the
lengths of the sides of the squares and ask how the volume Y(x) of the resulting
tank depends upon x. In particular, we want to know what x maximizes T'(x).
Show that
Y(x) = x(15 - 2x)(24 - 2x)
= 4x3 - 78x2 + 360x

and tell why x must be restricted to the interval 0 < x < -. Show that

Y'(x) = 12x2 - 156x + 360

=12(x-3)(x-10).

Tell why Y(x) is increasing when 0 < x < 3 and is decreasing when 3 < x <
Show that the maximum Y attainable is 486 cubic units.
10 A sheet-iron tank without a top is to have volume Y. A rectangular
sheet h feet high and 27rr feet long, costing 4 dollars per square foot, is bent and
welded into a circular cylinder to form the lateral surface of the tank. A sheet
2r feet square of different material, costing B dollars per square foot, is trimmed
to form a circular base which is welded to the cylinder to form the tank. Find
the radius and height of the tank for which the total cost T of the material (the
total amount purchased, not merely the amount actually used in the tank) is a
minimum. dns.:
sZE 1 a16B2Y
r = -B' h- r 142
Hint: Start by showing that
T = 2r.4rh + 4Br2

and then use the relation Y = rr2h to express T in terms of just one of the vari-

ables r and A. Standard methods may then be used to minimize T.

11 Referring to Problem 10, find the radius and height of the tank for w hich

the cost of the material actually used is a minimum.

12 Referring to Problems 10 and 11, find the radius and height which mini-

mize the cost of the material actually used in making

a tank which has a top exactly like the base.

291

Figure 5.292
13 A long rectangular sheet of tin is 2a inches
wide. Find the depth of the V-shaped trough of
maximum cross-sectional area (see Figure 5.292) that
can be made by bending the plate along its central
longitudinal axis. .4ns.: a/1/2.

14 A f 13 dF'^5293

a a

Figure 5.293

fter reerring to Problem an figure

MENME

formulate and solve a problem involving construction L

of troughs having rectangular cross sections.