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

Section 4.7 Optimization Problems 339
required to swim a distance L is Lysv 2 ud and the total
energy E required to swim the distance is given by
L
Esvd − av 3 ?
v 2 u
where a is the proportionality constant.
(a) Determine the value of v that minimizes E.
(b) Sketch the graph of E.
Note: This result has been verified experimentally; migrating
fish swim against a current at a speed 50% greater than
the current speed.
47. In a beehive, each cell is a regular hexagonal prism, open
at one end with a trihedral angle at the other end as in the
figure. It is believed that bees form their cells in such a
way as to min imize the surface area for a given side length
and height, thus using the least amount of wax in cell
construction. Examination of these cells has shown that
the measure of the apex angle is amazingly consistent.
Based on the geometry of the cell, it can be shown that the
surface area S is given by
S − 6sh 2 3 2 s2 cot 1 (3s 2 s3y2) csc
where s, the length of the sides of the hexagon, and h, the
height, are constants.
(a) Calculate dSyd.
(b) What angle should the bees prefer?
(c) Determine the minimum surface area of the cell (in
terms of s and h).
Note: Actual measurements of the angle in beehives
have been made, and the measures of these angles seldom
differ from the calculated value by more than 28.
rear
of cell
b
s
h
trihedral
angle ¨
front
of cell
48. A boat leaves a dock at 2:00 pm and travels due south at a
speed of 20 kmyh. Another boat has been heading due east
at 15 kmyh and reaches the same dock at 3:00 pm. At what
time were the two boats closest together?
49. Solve the problem in Example 4 if the river is 5 km wide
and point B is only 5 km downstream from A.
50. A woman at a point A on the shore of a circular lake with
radius 2 mi wants to arrive at the point C diametrically
;
opposite A on the other side of the lake in the shortest
possible time (see the figure). She can walk at the rate of
4 miyh and row a boat at 2 miyh. How should she proceed?
A
¨
2
51. An oil refinery is located on the north bank of a straight
river that is 2 km wide. A pipeline is to be constructed from
the refinery to storage tanks located on the south bank of
the river 6 km east of the refinery. The cost of laying pipe is
$400,000ykm over land to a point P on the north bank and
$800,000ykm under the river to the tanks. To minimize the
cost of the pipeline, where should P be located?
52. Suppose the refinery in Exercise 51 is located 1 km north
of the river. Where should P be located?
53. The illumination of an object by a light source is directly
proportional to the strength of the source and inversely proportional
to the square of the distance from the source.
If two light sources, one three times as strong as the other,
are placed 10 ft apart, where should an object be placed
on the line between the sources so as to receive the least
illumination?
54. Find an equation of the line through the point s3, 5d that cuts
off the least area from the first quadrant.
55. Let a and b be positive numbers. Find the length of the shortest
line segment that is cut off by the first quadrant and passes
through the point sa, bd.
56. At which points on the curve y − 1 1 40x 3 2 3x 5 does the
tangent line have the largest slope?
57. What is the shortest possible length of the line segment that
is cut off by the first quadrant and is tangent to the curve
y − 3yx at some point?
58. What is the smallest possible area of the triangle that is cut
off by the first quadrant and whose hypotenuse is tangent to
the parabola y − 4 2 x 2 at some point?
59. (a) If Csxd is the cost of producing x units of a commodity,
then the average cost per unit is csxd − Csxdyx. Show
that if the average cost is a minimum, then the marginal
cost equals the average cost.
(b) If Csxd − 16,000 1 200x 1 4x 3y2 , in dollars, find (i) the
cost, average cost, and marginal cost at a production level
of 1000 units; (ii) the production level that will minimize
the average cost; and (iii) the minimum average cost.
2
B
C
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