15. Cost A rectangular garden of area 75 square feet is to be sur- rounded on three sides by a brick wall costing $10 per foot and on one side by a fence costing $5 per foot. Find the dimensions of the garden that minimize the cost of materials. 16
15. Cost A rectangular garden of area 75 square feet is to be sur- rounded on three sides by a brick wall costing $10 per foot and on one side by a fence costing $5 per foot. Find the dimensions of the garden that minimize the cost of materials. 16
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exc 2.5
Q15 and Q19 needed to be solved correctly in 30 minutes and get the thumbs up please show me neat and clean work for it
![15. Cost A rectangular garden of area 75 square feet is to be sur-
rounded on three sides by a brick wall costing S10 per foot
and on one side by a fence costing $5 per foot. Find the
dimensions of the garden that minimize the cost of materials.
16. Cost A closed rectangular box with a square base and a vol-
ume of 12 cubic feet is to be constructed from two different
types of materials. The top is made of a metal costing $2 per
square foot and the remainder of wood costing S1 per square
foot. Find the dimensions of the box for which the cost of
materials is minimized.
17. Surface Area Find the dimensions of the closed rectangular
box with square base and volume 8000 cubic centimeters that
can be constructed with the least amount of material.
18. Volume A canvas wind shelter for the beach has a back, two
square sides, and a top. Find the dimensions for which the
volume will be 250 cubic feet and that requires the least pos-
sible amount of canvas.
19. Area A farmer has $1500 available to build an E-shaped fence
along a straight river so as to create two identical rectangu-
lar pastures. (See Fig. 13.) The materials for the side parallel
to the river cost $6 per foot, and the materials for the three
sections perpendicular to the river cost $5 per foot. Find the
dimensions for which the total area is as large as possible.
d
Figure 13 Rectangular
pastures along a river.
20. Area Find the dimensions of the rectangular garden of great-
est area that can be fenced off (all four sides) with 300 meters
of fencing.
21. Maximizing a Product Find two positive numbers, x and y,
whose sum is 100 and whose product is as large as possible.
22. Minimizing a Sum Find two positive numbers, x and y, whose
product is 100 and whose sum is as small as possible.
River
23. Area Figure 14(a) shows a Norman window, which consists
of a rectangle capped by a semicircular region. Find the value
of x such that the perimeter of the window will be 14 feet and
the area of the window will be as large as possible.
2πχ
h
h
(b)
Side unrolled
2x
(a)
Figure 14
maldor
24. Surface Area A large soup can is to be designed so that the
can will hold 167 cubic inches (about 28 ounces) of soup. [See
Fig. 14(b).] Find the values of x and h for which the amount
of metal needed is as small as possible.
h
25. In Example 3 we can solve the constraint equation (2) for x instead
of w to get x = 20-w. Substituting this for x in (1), we get
A = xw = 20 --ww.
-1/w)w.
2.5 Optimization Problems 169
Sketch the graph of the equation
A = 20w -².
and show that the maximum occurs when w= 20 and x 10.
26. Cost A ship uses 5x² dollars of fuel per hour when traveling at
a speed of x miles per hour. The other expenses of operating
the ship amount to $2000 per hour. What speed minimizes the
cost of a 500-mile trip? (Hint: Express cost in terms of speed
and time. The constraint equation is distance speed x time.]
27. Cost A cable is to be installed from one corner, C, of a rectan-
gular factory to a machine, M, on the floor. The cable will run
along one edge of the floor from C to a point, P, at a cost of
$6 per meter, and then from P to M in a straight line buried
under the floor at a cost of $10 per meter (see Fig. 15). Let x
denote the distance from C to P. If M is 24 meters from the
nearest point, A, on the edge of the floor on which lies, and
A is 20 meters from C, find the value of x that minimizes the
cost of installing the cable and determine the minimum cost.
20
P
Figure 16
24
Figure 15
28. Area A rectangular page is to contain 50 square inches of print.
The page has to have a 1-inch margin on top and at the bottom
and a -inch margin on each side (see Fig. 16). Find the dimen-
sions of the page that minimize the amount of paper used.
in.
in.
y
Print Area Popt Aree Print Ares
Print Area Prist Area Print Area
Print Area Print Area Print Area
Prist Aron Print Area Print Area
Print Area
M
1 in.
1 in.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ac7d000-e164-4b22-a743-e43daa5d484c%2Ff4bd21b3-f9d3-43b9-8fa3-6ba3b102e9af%2F5ye03j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:15. Cost A rectangular garden of area 75 square feet is to be sur-
rounded on three sides by a brick wall costing S10 per foot
and on one side by a fence costing $5 per foot. Find the
dimensions of the garden that minimize the cost of materials.
16. Cost A closed rectangular box with a square base and a vol-
ume of 12 cubic feet is to be constructed from two different
types of materials. The top is made of a metal costing $2 per
square foot and the remainder of wood costing S1 per square
foot. Find the dimensions of the box for which the cost of
materials is minimized.
17. Surface Area Find the dimensions of the closed rectangular
box with square base and volume 8000 cubic centimeters that
can be constructed with the least amount of material.
18. Volume A canvas wind shelter for the beach has a back, two
square sides, and a top. Find the dimensions for which the
volume will be 250 cubic feet and that requires the least pos-
sible amount of canvas.
19. Area A farmer has $1500 available to build an E-shaped fence
along a straight river so as to create two identical rectangu-
lar pastures. (See Fig. 13.) The materials for the side parallel
to the river cost $6 per foot, and the materials for the three
sections perpendicular to the river cost $5 per foot. Find the
dimensions for which the total area is as large as possible.
d
Figure 13 Rectangular
pastures along a river.
20. Area Find the dimensions of the rectangular garden of great-
est area that can be fenced off (all four sides) with 300 meters
of fencing.
21. Maximizing a Product Find two positive numbers, x and y,
whose sum is 100 and whose product is as large as possible.
22. Minimizing a Sum Find two positive numbers, x and y, whose
product is 100 and whose sum is as small as possible.
River
23. Area Figure 14(a) shows a Norman window, which consists
of a rectangle capped by a semicircular region. Find the value
of x such that the perimeter of the window will be 14 feet and
the area of the window will be as large as possible.
2πχ
h
h
(b)
Side unrolled
2x
(a)
Figure 14
maldor
24. Surface Area A large soup can is to be designed so that the
can will hold 167 cubic inches (about 28 ounces) of soup. [See
Fig. 14(b).] Find the values of x and h for which the amount
of metal needed is as small as possible.
h
25. In Example 3 we can solve the constraint equation (2) for x instead
of w to get x = 20-w. Substituting this for x in (1), we get
A = xw = 20 --ww.
-1/w)w.
2.5 Optimization Problems 169
Sketch the graph of the equation
A = 20w -².
and show that the maximum occurs when w= 20 and x 10.
26. Cost A ship uses 5x² dollars of fuel per hour when traveling at
a speed of x miles per hour. The other expenses of operating
the ship amount to $2000 per hour. What speed minimizes the
cost of a 500-mile trip? (Hint: Express cost in terms of speed
and time. The constraint equation is distance speed x time.]
27. Cost A cable is to be installed from one corner, C, of a rectan-
gular factory to a machine, M, on the floor. The cable will run
along one edge of the floor from C to a point, P, at a cost of
$6 per meter, and then from P to M in a straight line buried
under the floor at a cost of $10 per meter (see Fig. 15). Let x
denote the distance from C to P. If M is 24 meters from the
nearest point, A, on the edge of the floor on which lies, and
A is 20 meters from C, find the value of x that minimizes the
cost of installing the cable and determine the minimum cost.
20
P
Figure 16
24
Figure 15
28. Area A rectangular page is to contain 50 square inches of print.
The page has to have a 1-inch margin on top and at the bottom
and a -inch margin on each side (see Fig. 16). Find the dimen-
sions of the page that minimize the amount of paper used.
in.
in.
y
Print Area Popt Aree Print Ares
Print Area Prist Area Print Area
Print Area Print Area Print Area
Prist Aron Print Area Print Area
Print Area
M
1 in.
1 in.
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