5. Suppose the objective function P = xy is subject to the constraint 10x + y = 100, where x and y are real numbers. 19. Minimum-surface-area box Of all boxes with a square base and a volume of 8 m, which one has the minimum surface area? (Give its dimensions.) a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x. b. Find the absolute maximum value of P subject to the given 20. Maximum-volume box Suppose an airline policy states that all baggage must be box-shaped with a sum of length, width, and height not exceeding 108 in. What are the dimensions and volume of a square-based box with the greatest volume under these condi tions? constraint. Suppose S x + 2y is an objective function subject to the con- straint xy = 50, for x > 0 and y > 0. 6. 21. Shipping crates A square-based, box-shaped shipping crate is a. Eliminate the variable y from the objective function so that S is expressed as a function of one variable x. b. Find the absolute minimum value of S subject to the given constraint. designed to have a volume of 16 ft. The material used to make the base costs twice as much (per square foot) as the material in the sides, and the material used to make the top costs half as much (per square foot) as the material in the sides. What are the dimen- 7. What two nonnegative real numbers with a sum of 23 have the sions of the crate that minimize the cost of materials? largest possible product? 22. Closest point on a line What point on the line y = 3x + 4 is closest to the origin? 8. What two nonnegative real numbers a and b whose sum is 23 maximize a + b? Minimize a? + b? What two positive real numbers whose product is 50 have the smallest possible sum? 23. Closest point on a curve What point on the parabola y = 1 – x² is closest to the point (1, 1)? 9. 10. Find numbers x and y satisfying the equation 3x + y = 12 such that the product of x and y is as large as possible. 24. Minimum distance Find the point P on the curve y = x that is closest to the point (18, 0). What is the least distance between P and (18, 0)? (Hint: Use synthetic division.) %3D 25. Minimum distance Find the point P on the line y = 3x that is closest to the point (50, 0). What is the least distance between P and (50, 0)? Practice Exercises 11. Maximum-area rectangles Of all rectangles with a perimeter of 10, which one has the maximum area? (Give the dimensions.) 12. Maximum-area rectangles Of all rectangles with a fixed perim- eter of P, which one has the maximum area? (Give the dimensions in terms of P.) 26. Walking and swimming A man wishes to get from an initial point on the shore of a circular pond with radius 1 mi to a point on the shore directly opposite (on the other end of the diameter). He plans to swim from the initial point to another point on the shore 13. Minimum-perimeter rectangles Of all rectangles of area 100, which one has the minimum perimeter? and then walk along the shore to the terminal point. 14. Minimum-perimeter rectangles Of all rectangles with a fixed area A, which one has the minimum perimeter? (Give the dimen- sions in terms of A.) a. If he swims at 2 mi/hr and walks at 4 mi/hr, what are the maximum and minimum times for the trip? b. If he swims at 2 mi/hr and walks at 1.5 mi/hr, what are the maximum and minimum times for the trip? c. If he swims at 2 mi/hr, what is the minimum walking speed for which it is quickest to walk the entire distance? 15. Minimum sum Find positive numbers x and y satisfying the equa- tion xy = 12 such that the sum 2x + y is as small as possible. 27. Walking and rowing A boat on the ocean is 4 mi from the nearest point on a straight shoreline; that point is 6 mi from a restaurant on the shore (see figure). A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. 16. Pen problems a. A rectangular pen is built with one side against a barn. Two hundred meters of fencing are used for the other three sides of the pen. What dimensions maximize the area of the pen? b. A rancher plans to make four identical and adjacent rectan- gular pens against a barn, each with an area of 100 m² (see figure). What are the dimensions of each pen that minimize the a. If she walks at 3 mi/hr and rows at 2 mì/hr, at which point on the shore should she land to mìnimize the total travel time? b. If she walks at 3 mi/hr, what is the minimum speed at which she must row so that the quickest way to the restaurant is to row directly (with no walking)? amount of fence that must be used? Barn 100 100 100 100 4 mi 21,981 10 APR PAGES 15 TLO 3D MacBook Pro
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
could you help with 21, I'm doing random problems on my own for fun, got stuck on this one
![5. Suppose the objective function P = xy is subject to the constraint
10x + y = 100, where x and y are real numbers.
19. Minimum-surface-area box Of all boxes with a square base
and a volume of 8 m, which one has the minimum surface area?
(Give its dimensions.)
a. Eliminate the variable y from the objective function so that P is
expressed as a function of one variable x.
b. Find the absolute maximum value of P subject to the given
20. Maximum-volume box Suppose an airline policy states that all
baggage must be box-shaped with a sum of length, width, and
height not exceeding 108 in. What are the dimensions and volume
of a square-based box with the greatest volume under these condi
tions?
constraint.
Suppose S x + 2y is an objective function subject to the con-
straint xy = 50, for x > 0 and y > 0.
6.
21.
Shipping crates A square-based, box-shaped shipping crate is
a. Eliminate the variable y from the objective function so that S is
expressed as a function of one variable x.
b. Find the absolute minimum value of S subject to the given
constraint.
designed to have a volume of 16 ft. The material used to make
the base costs twice as much (per square foot) as the material in
the sides, and the material used to make the top costs half as much
(per square foot) as the material in the sides. What are the dimen-
7.
What two nonnegative real numbers with a sum of 23 have the
sions of the crate that minimize the cost of materials?
largest possible product?
22. Closest point on a line What point on the line y = 3x + 4 is
closest to the origin?
8.
What two nonnegative real numbers a and b whose sum is 23
maximize a + b? Minimize a? + b?
What two positive real numbers whose product is 50 have the
smallest possible sum?
23. Closest point on a curve What point on the parabola y = 1 – x²
is closest to the point (1, 1)?
9.
10. Find numbers x and y satisfying the equation 3x + y = 12 such
that the product of x and y is as large as possible.
24. Minimum distance Find the point P on the curve y = x that is
closest to the point (18, 0). What is the least distance between P
and (18, 0)? (Hint: Use synthetic division.)
%3D
25. Minimum distance Find the point P on the line y = 3x that is
closest to the point (50, 0). What is the least distance between P
and (50, 0)?
Practice Exercises
11. Maximum-area rectangles Of all rectangles with a perimeter of
10, which one has the maximum area? (Give the dimensions.)
12. Maximum-area rectangles Of all rectangles with a fixed perim-
eter of P, which one has the maximum area? (Give the dimensions
in terms of P.)
26. Walking and swimming A man wishes to get from an initial
point on the shore of a circular pond with radius 1 mi to a point on
the shore directly opposite (on the other end of the diameter). He
plans to swim from the initial point to another point on the shore
13. Minimum-perimeter rectangles Of all rectangles of area 100,
which one has the minimum perimeter?
and then walk along the shore to the terminal point.
14. Minimum-perimeter rectangles Of all rectangles with a fixed
area A, which one has the minimum perimeter? (Give the dimen-
sions in terms of A.)
a. If he swims at 2 mi/hr and walks at 4 mi/hr, what are the
maximum and minimum times for the trip?
b. If he swims at 2 mi/hr and walks at 1.5 mi/hr, what are the
maximum and minimum times for the trip?
c. If he swims at 2 mi/hr, what is the minimum walking speed
for which it is quickest to walk the entire distance?
15. Minimum sum Find positive numbers x and y satisfying the equa-
tion xy = 12 such that the sum 2x + y is as small as possible.
27. Walking and rowing A boat on the ocean is 4 mi from the nearest
point on a straight shoreline; that point is 6 mi from a restaurant on
the shore (see figure). A woman plans to row the boat straight to a
point on the shore and then walk along the shore to the restaurant.
16. Pen problems
a. A rectangular pen is built with one side against a barn. Two
hundred meters of fencing are used for the other three sides of
the pen. What dimensions maximize the area of the pen?
b. A rancher plans to make four identical and adjacent rectan-
gular pens against a barn, each with an area of 100 m² (see
figure). What are the dimensions of each pen that minimize the
a. If she walks at 3 mi/hr and rows at 2 mì/hr, at which point on
the shore should she land to mìnimize the total travel time?
b. If she walks at 3 mi/hr, what is the minimum speed at which
she must row so that the quickest way to the restaurant is to
row directly (with no walking)?
amount of fence that must be used?
Barn
100
100
100
100
4 mi
21,981
10
APR
PAGES
15
TLO
3D
MacBook Pro](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffef29e8a-9cde-4dab-a8ca-f9b40140f207%2F845638b9-3c52-42fa-a802-fea8d61a3a04%2F13cclaa.jpeg&w=3840&q=75)
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