F. Optimization with Constraint Find the maximum of Q = xy if x+y = 2. Optimisatio Tak
F. Optimization with Constraint Find the maximum of Q = xy if x+y = 2. Optimisatio Tak
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Exc 2.5
Q5 needed
![EXERCISES 2.5
1. For what x does the function g(x) = 10 + 40x-x² have its
maximum value? bail c tow
2. Find the maximum value of the function f(x) = 12x - x²,
and give the value of x where this maximum occurs.
3. Find the minimum value of f(t) = 1³-61² + 40, 1 ≥ 0, and
give the value of where this minimum occurs.
4. For what does the function f(t) = 1²-241 have its mini-
mum value?
5. Optimization with Constraint Find the maximum of Q = xy if
x+y = 2.
6. Optimization with Constraint Find two positive numbers x and
y that maximize Q=x²y if x + y = 2.
7. Optimization with Constraint Find the minimum of
Q=x² + y²if x+y = 6.
8. In Exercise 7, can there be a maximum for Q = x² + y² if
x + y = 6? Justify your answer.
9. Minimizing a Sum Find the positive values of x and y that
minimize S = x + y if xy = 36, and find this minimum value.
10. Maximizing a Product Find the positive values of x, y, and z
that maximize Q = xyz, if x + y = 1 and y + z = 2. What is
this maximum value?
(b) Express the quantity to be maximized as a function of x.
(c) Find the optimal values of x and y.
12. Volume Figure 12(b) shows an open rectangular box with a
square base. Consider the problem of finding the values of
x and h for which the volume is 32 cubic feet and the total
surface area of the box is minimal. (The surface area is the
sum of the areas of the five faces the box.)
(a) Determine the objective and constraint equations.
(b) Express the quantity to be minimized as a function of x.
(c) Find the optimal values of x and h.
11. Area There are $320 available to fence in a rectangular gar-d(e)
den. The fencing for the side of the garden facing the road
costs $6 per foot and the fencing for the other three sides costs.
S2 per foot. [See Fig. 12(a).] Consider the problem of finding
the dimensions of the largest possible garden.
(a) Determine the objective and constraint equations.
$2
Figure 11 Wind shelter.
Road
(a)
h
(b)
Figure 12
13. Volume Postal requirements specify that parcels must have
length plus girth of at most 84 inches. Consider the problem
of finding the dimensions of the square-ended rectangular
package of greatest volume that is mailable.
(a) Draw a square-ended rectangular box. Label each edge of
the square end with the letter x and label the remaining
dimension of the box with the letter h.
(b) Express the length plus the girth in terms of x and h.
Determine the objective and constraint equations.
(d) Express the quantity to be maximized as a function of x.
(e) Find the optimal values of x and h.
14. Perimeter Consider the problem of finding the dimensions of
the rectangular garden of area 100 square meters for which
the amount of fencing needed to surround the garden is as
small as possible.
(a) Draw a picture of a rectangle and select appropriate let-
ters for the dimensions.
(b) Determine the objective and constraint equations.
(c) Find the optimal values for the dimensions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ac7d000-e164-4b22-a743-e43daa5d484c%2F482165cf-ccbf-4d8b-a9e5-ad1c8c39f7f4%2Fq3qp28b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:EXERCISES 2.5
1. For what x does the function g(x) = 10 + 40x-x² have its
maximum value? bail c tow
2. Find the maximum value of the function f(x) = 12x - x²,
and give the value of x where this maximum occurs.
3. Find the minimum value of f(t) = 1³-61² + 40, 1 ≥ 0, and
give the value of where this minimum occurs.
4. For what does the function f(t) = 1²-241 have its mini-
mum value?
5. Optimization with Constraint Find the maximum of Q = xy if
x+y = 2.
6. Optimization with Constraint Find two positive numbers x and
y that maximize Q=x²y if x + y = 2.
7. Optimization with Constraint Find the minimum of
Q=x² + y²if x+y = 6.
8. In Exercise 7, can there be a maximum for Q = x² + y² if
x + y = 6? Justify your answer.
9. Minimizing a Sum Find the positive values of x and y that
minimize S = x + y if xy = 36, and find this minimum value.
10. Maximizing a Product Find the positive values of x, y, and z
that maximize Q = xyz, if x + y = 1 and y + z = 2. What is
this maximum value?
(b) Express the quantity to be maximized as a function of x.
(c) Find the optimal values of x and y.
12. Volume Figure 12(b) shows an open rectangular box with a
square base. Consider the problem of finding the values of
x and h for which the volume is 32 cubic feet and the total
surface area of the box is minimal. (The surface area is the
sum of the areas of the five faces the box.)
(a) Determine the objective and constraint equations.
(b) Express the quantity to be minimized as a function of x.
(c) Find the optimal values of x and h.
11. Area There are $320 available to fence in a rectangular gar-d(e)
den. The fencing for the side of the garden facing the road
costs $6 per foot and the fencing for the other three sides costs.
S2 per foot. [See Fig. 12(a).] Consider the problem of finding
the dimensions of the largest possible garden.
(a) Determine the objective and constraint equations.
$2
Figure 11 Wind shelter.
Road
(a)
h
(b)
Figure 12
13. Volume Postal requirements specify that parcels must have
length plus girth of at most 84 inches. Consider the problem
of finding the dimensions of the square-ended rectangular
package of greatest volume that is mailable.
(a) Draw a square-ended rectangular box. Label each edge of
the square end with the letter x and label the remaining
dimension of the box with the letter h.
(b) Express the length plus the girth in terms of x and h.
Determine the objective and constraint equations.
(d) Express the quantity to be maximized as a function of x.
(e) Find the optimal values of x and h.
14. Perimeter Consider the problem of finding the dimensions of
the rectangular garden of area 100 square meters for which
the amount of fencing needed to surround the garden is as
small as possible.
(a) Draw a picture of a rectangle and select appropriate let-
ters for the dimensions.
(b) Determine the objective and constraint equations.
(c) Find the optimal values for the dimensions.
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