5.2 Trends, maxima, and minima 301hours per week. Find the number of secretaries that will turnout the most
work. Discussion and solution: If there are x secretaries, the number y(x) of
hours of work done per week is
x2\l x3(1) y(x) =x(30- 0) = 30x- 30
It is required that x be an integer in the interval 0 <- x 5 30, so there are only
31 possibilities. We can calculate the 31 numbers y(0), y(1), , y(30) and
select the x which gives the greatest y(x). It is easier and more enlightening,
however, to use some calculus. Forgetting momentarily that x is an integer
number of secretaries, we observe that (1) defines y(x) for each real x. Differ-
entiating gives
X2 1
(2) y'(x)=30-To -10(300-x'-).
Thus y'(x) > 0 and y is increasing when 0 5 x < V/3-o(). Moreover, y'(x) < 0
and y is decreasing when x > 300. Since 300 = 17.32, we see that y(x) <
y(17) when 0 < x < 17 and that y(x) < y(18) when 18 < x =< 30. Thus the
answer is 17 if y(17) > y(18) and is 18 if y(18) > y(17).
22 As in Figure 5.295, a triangle is inscribed in a semicircular
region having diameter a. Find the 0 which maximizes the area
of the triangle. A'ns.: 0 = 7r/4. a
23 A printed page is required to contain -4 square units of Figure 5.295
printed matter. Side margins of widths a and top and bottom
margins of widths b are required. Find the lengths of the printed lines when
the page is designed to use the least paper. .4ns.: ,/afl/b
24 Sketch a reasonably good graph of y = x22 and then mark the point or
points on this graph that seem to be closest to the point (0,1). Then calculate
the coordinates of the closest point or points. Hint: Minimize the square of the
distance from the point (0,1) to the point (x,x2).
25 The strength (ability to resist bending) of a rectangular beam is propor-
tional to the width x and to the square of the height y of a cross section. Find
the width and height of the strongest beam
that can be sawed from a cylindrical log
whose cross sections are circular disks
of diameter L. .4ns.: Width = L/V,
height = N/2-L/-.
26 The x axis of Figure 5.296 is the
southern shore of a lake containing a little
island at the point (a,b), where a > 0. A 0
man who is at the origin can run r feet per
second along the x axis and can swim s feet
Run x
Figure 5.296per second in the water. He wants to reach the island as quickly as possible.
Should he do some running before he starts to swim and, if so, how far? Partial
ans.: He should run
a r2-s2b