James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

362 Chapter 4 Applications of Differentiation
beam. Determine the dimensions of the planks that will
have maximal cross-sectional area.
(c) Suppose that the strength of a rectangular beam is
proportional to the product of its width and the square of
its depth. Find the dimensions of the strongest beam that
can be cut from the cylindrical log.
80. If a projectile is fired with an initial velocity v at an angle
of inclination from the horizontal, then its trajectory,
neglecting air resistance, is the parabola
t
y − stan dx 2
2v 2 cos 2 x 2 0 , , 2
(a) Suppose the projectile is fired from the base of a plane
that is inclined at an angle , . 0, from the horizontal,
as shown in the figure. Show that the range of the projectile,
measured up the slope, is given by
Rsd − 2v 2 cos sins 2 d
t cos 2
(b) Determine so that R is a maximum.
(c) Suppose the plane is at an angle below the horizontal.
Determine the range R in this case, and determine the
angle at which the projectile should be fired to maximize
R.
82. If a metal ball with mass m is projected in water and the force
of resistance is proportional to the square of the velocity, then
the distance the ball travels in time t is
sstd − m c
ln coshÎ
tc
mt
where c is a positive constant. Find lim c l 0 1 sstd.
83. Show that, for x . 0,
x
1 1 x 2 , tan21 x , x
84. Sketch the graph of a function f such that f 9sxd , 0 for
all x, f 0sxd . 0 for | x | . 1, f 0sxd , 0 for | x | , 1, and
lim x l6` f f sxd 1 xg − 0.
85. A light is to be placed atop a pole of height h feet to
illuminate a busy traffic circle, which has a radius of 40 ft.
The intensity of illumination I at any point P on the circle
is directly proportional to the cosine of the angle (see the
figure) and inversely proportional to the square of the distance
d from the source.
(a) How tall should the light pole be to maximize I?
(b) Suppose that the light pole is h feet tall and that a
woman is walking away from the base of the pole at the
rate of 4 ftys. At what rate is the intensity of the light at
the point on her back 4 ft above the ground decreasing
when she reaches the outer edge of the traffic circle?
y
h
¨
d
0
¨
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81. If an electrostatic field E acts on a liquid or a gaseous polar
dielectric, the net dipole moment P per unit volume is
PsEd − e E 1 e 2E
e E 2 e 2E 2 1 E
Show that lim E l 0 1 PsEd − 0.
x
86. Water is flowing at a constant rate into a spherical tank. Let
Vstd be the volume of water in the tank and Hstd be the height
of the water in the tank at time t.
(a) What are the meanings of V9std and H9std? Are these
derivatives positive, negative, or zero?
(b) Is V 0std positive, negative, or zero? Explain.
(c) Let t 1, t 2, and t 3 be the times when the tank is one-quarter
full, half full, and three-quarters full, respectively.
Are the values H0st 1d, H0st 2d, and H0st 3d positive,
negative, or zero? Why?
40
P
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