3•
3. The product is 192 and the sum is a minimum.
4. The product is 192 and the sum of the first plus three times the
second is a minimum.
5. The second number is the reciprocal of the first and the sum is
a minimum.
6. The sum of the first and twice the second is 100 and the product
is a maximum.
17. Area A farmer plans to fence a rectangular pasture
adjacem
to a riser. The pasture must contain 180,000 square
meters
in order to provide enough grass for the herd. What dimen
sions would ruire the least amount of fencing if no fencing is
needed along th iiver?
18. Area A rancher has 200 feet of tencinc with which to enclose
t o ad jacent rect an eniar corral
e ticure What d eden cions
should be used so tan the enclosed i a siN he a
In Exercises 7 and 8, find the length and width of a rectangle
that has the given perimeter and a maximum area.
7. Perimeter: 100 meters
8. Perimeter: P units
In Exercises 9 and 10, find the length and width of a rectangle
that has the given area and a minimum perimeter.
S
—
r
_
9, Area: 64 square feet
19, Volume
10. Area: A square centimeters
In Exercises 11—14, find the point on the graph of the function
that is closest to the given point.
Ii. f(x(
=
13. fed
(4, 0)
12. fed
=
(2*
14. fix)
=
SECTION 3.7
Point
Function_-
Pomi
x— 8
0
(2. 0)
(x ± 1
(5. 3)
Optimization Problems
(a) Verif that each of the rectangular solids shown in the
figure has a surface area of 150 square inches.
(b) Find the volume of each.
(c) Determine the dimensions of a rectangular solid (with a
square base) of maximum volume if its surface area is i)
square inches.
.
.
217
23. Area A Norman window is constructed by adjoining a
semi
circle to the top of an ordinary rectangular window (see figure).
Find the dimensions of a Norman window of maximum area if
the total perimeter is 16 feet.
Figure for 27
27. Area A_rectangle is bounded by the x-axis and the semicircle
v = /25
2 (see figure). What length and width should the
x
rectangle have so that its area is a maximum?
—
use
ii
s.—.____-- x
41
—
24. Area A rectangle is bounded by the x- and v-axes and the
graph of v
(6
x)/2 (see figure). What length and width
should the rectangle have so that its area is a maximum?
—
y
A
y
4
4-i
6r
1’
56
1
Figure for 24
2
Figure for 25
3
4
28. Area Find the dimensions
of the largest rectangle that
can be
inscribed in a semicircle of radius i(see Exercise 2?)
29. Area A rectangul page
is to contain 30 square
inches of
print. The margins on each side
are 1 inch. Find the dimensions
of the page sued that the least
amount of paper is used.
30. Area A reunguIar page
is to contain 36 square
inches of pinit.
The margins on each side are
to be 1 inches Find the
dpien
siops of ted o e cued that
the least amount of naper is
used
20 \ umencal Grapha4 and
Analy& Anals,s n
open box
of marmnurn volume is to
be made from a square
piece of mate
rial. 24 inches on a side, by
Cutting equal squares from
the cor
ners and turning up the sides
()
Use calculus to find
the critical number of
the function
and find the maximum
value.