What is Optimization: Definition and 627 Discussions
Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains.
Homework Statement
Consider the part of the parabola y=1-x^2 from x=-1 to x=1. This curve fits snugly inside an isosceles triangle with base on the x-axis and one vertex on the y-axis. What is the smallest possible area of such a triangle?
The Attempt at a Solution
A =...
My job is to maximize the area of the kite so that it will fly better, faster, higher. A kite frame is to be made from six pieces of wood. The four pieces that form its border have been cut the lengths. So one of the top border is 2 cm long and the bottom border is 4 cm. The total border length...
Hey all,
I'm struggling to even start this problem:
An evaporative cooler design uses a rotating wheel that is placed upright (vertical) and is partially submerged in water. The center of the circle is above the waterline. Let R be the radius of the wheel and x be the distance from the...
I know how to optimize, but I'm having trouble starting both. I think the trick to the first one is something to do with similar triangles, but I can't quite articulate an equation to do the problem with. I know the second one requires the volume, and I think the area of a circle or the surface...
This is how the book describes the problem:
If the ellipse x2/a2+y2/b2=1 is to enclose the circle x2+y2=2y, what values of a and b minimize the are of the ellipse?
First of all I completed the square for the second equation and I got: x2+(y-1)2=1. I isolated the x2 and substituted it into...
Homework Statement
"For this project we locate a trash dumpster in order to study its shape and constuction. We then attempt to determine the dimensions of a container of similar design that minimize construction cost."
1. (Already located, measured, and descibed a dumpster found).
2...
Hello everyone,
My questions and link to my webpage is posted over at the original science forums. Please take a look:
http://www.scienceforums.net/forum/showthread.php?t=36161
Cheers,
Michael
Homework Statement
look at jpg attachment
Homework Equations
T(y)=(z-y/r)+(sqrt(x^2+y^2)/s)
ac=z
bc=x
dc=y
ab=w
im having trouble taking the derivative of T(y) and how to solve it
on the second one i think there is no maximal area but there is a minimal but not sure how to...
Homework Statement
A water tank is in the shape of an inverted conical cone with top radius of 20m and
depth of 15m. Water is flowing into the tank at a rate of 0.1m^3/min.
(a) How fast is the depth of water in the tank increasing when the depth is 5m?
Water is now leaking from the...
Greetings,
I'm working on a problem where I am to find the coordinates of the point (x,y,z) to the plane z=3x+2y+1, which is closest to the origin.
I know that this is an optimization problem, and I believe I have to minimize (x,y,3x+2y+1).
I started by finding partial derivative, fx, of the...
Hi,
This is a question about a boolean "law" type behavior I've noticed in my homework a couple of times.
Basically i can't find a boolean algebra law that permits this optimization short of using a k-map.
So I'm just wondering if theirs some way to optimize the one equation using...
Homework Statement
This isn't that hard but I cannot remember a nice Calculus way of doing it. I'm trying to find the ratio of height to diameter of a cylinder that produces the minimum material buckling (B_m)^2. The problem statement my professor provided states that the minimum is found at...
I met a problem about finding the optimization of some function. I used the Trust-Region Newton and Quasi-Newton methods for the problem; however, with different initial guesses I sometimes got the local minimums. May I ask how to get out the trap of the local minimums please?
I may try the...
I met a problem about finding the optimization of some function. I used the Trust-Region Newton and Quasi-Newton methods for the problem; however, with different initial guesses I sometimes got the local minimums. May I ask how to get out the trap of the local minimums please?
I may try the...
a chord AB of a circle subtends an angle that is not equal to 60 degees at a point C on the circumference. ABC has maximum area. then find A & B in terms of the angle.
Hi. I have a problem I am hoping you all can shed some light on.
I have N entities, O, each described by N values - a weight W and N-1 similarity coefficients to the other N-1 entities. I guess we can represent Oi as (Wi, Sij, j=(1,2,...,N, i!=j)(?).
Given an integer M and M < N I need to...
Homework Statement
I need help on an optimization problem involving a hexagonal prism with no bottom or top, but the top is covered by a trihedral pyramid which has a displacement, x, such that the surface area of the object is at a minimum for a given volume. The assigned variables include...
I'm trying to find the regular parallelepiped with sides parallel to the coordinate axis inscribed in the ellipsoid x[2]/a[2] + y[2]/b[2] + z[2]/c[2] = 1 that has the largest volume. I've been trying the Lagrangian method: minimize f = (x)(y)(z), subject to the constraint (x[2]/a[2] + y[2]/b[2]...
Dear all,
I am looking for some advise on optimization routines. I have a collection of 2D data (x-y plot) and a piece of code which generates different models based upon several inputs (a,b,c,d,etc). These inputs generate several outputs which characterize the final generated model...
I have a case where I am trying to find the optimal place to put 3 rivet points (I will maybe even test with 4). The rivet points r subject to a force according to the picture, which vary 360. What I am seeking here is mathematical methods to find the optimal place to put the 3 riverts, so that...
Homework Statement
A train leaves the station at 10:00 and travels due south at a speed of 60 km/h. Another train has been heading due west at 45 km/h and reaches the same station at 11:00. At what time were the two trains closest together?
Homework Equations
c^{2}=a^{2}+b^{2}The Attempt at a...
[SOLVED] Optimization: Rectangle Inscribed in Triangle
Homework Statement
Please see http://www.jstor.org/pss/2686484 link. The problem I have is pretty much exactly the same as that dealt with in this excerpt.
(focus on the bit with the heading "What is the biggest rectangle you can...
[SOLVED] Optimization problem
Homework Statement
A baseball team plays in the stadium that holds 60000 spectators. With the ticket price at 8 the average attendence has been 24000. When the price dropped to 7, the average attendence rose to 30,000.
a) find the demand function p(x), where x...
Homework Statement
An eight-foot fence stands on level ground is one foot from a telephone pole. Find the shortest ladder that will reach over the fence to the pole.
Homework Equations
Pythagoras?
Derivative.
The Attempt at a Solution
The problem is I don't know how to start this...
1) http://www.geocities.com/asdfasdf23135/advcal28.JPG
From the assumptions, I think that the mean value theorem and/or the extreme value theorem may be helpful in this problem, but I can't figure out how to apply them to reach the conclusion. Could someone please give me some general hints...
Hello!
Here's a question that I couldn't understand;
A canvas tent is to be constructed in the shape of a right-circular cone with the ground as base;
Using the volume V and curved surface area S of the cone,
V = \frac{1}{3}\pi r^2 h, S = \pi rl,
Find the dimensions of the cone...
1a) Determine the maximum value of f(x,y,z)=(xyz)1/3 given that x,y,z are nonnegative numbers and x+y+z=k, k a constant.
1b) Use the result in (a) to show that if x,y,z are nonnegative numbers, then (xyz)1/3 < (x+y+z)/3
Attempt:
1a) Using the Lagrange Multiplier method, I get that the...
[SOLVED] one last optimization problems
Homework Statement
find two positive numbers such that the sum of the number and its reciprocal is as small as possible.
Homework Equations
x+(1/x) = s
f' =1 + ln x
The Attempt at a Solution
lost
Homework Statement
find the dimensions of a rectangle with perimeter 100m whose area is as large as possible
Homework Equations
area = XY
100 = 2x + 2y
y= 100/4x
x(100/4x)
(400x - 400x)/16x^2
1/16x^2 = 0
The Attempt at a Solution
well...
am lost
[SOLVED] simple optimization problem
Homework Statement
find the dimensions of a rectangle with area 1000 m^2 whose perimeter is as small as possible
Homework Equations
perimeter = 2x + 2y
area = xy
1000 = xy
y= 1000/x
perimeter = 2x + 2(1000/x)
The Attempt at a...
Homework Statement
A hardware store sells approximately 10 000 light bulbs a year. The owner wishes to determine how large an inventory of x (thousand) bulbs should be kept to minimize the cost for inventory. The carrying cost for the bulbs is $40/1000 while the paperwork for ordering is $12...
[SOLVED] Optimization Problem
Homework Statement
A tin can is to have a given capacity. Find the ratio of height to diameter if the amount of tin (total surface area) is a minimum.
Homework Equations
c=pi(r^2)h
surface area = 2pi(r^2)+2h(pi)r
The Attempt at a Solution
h=...
[SOLVED] Optimization problem
Homework Statement
A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the smicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 28 feet?
Homework...
Calculus: Optimization Problems
Homework Statement
Find the area of the largest rectangle that can be inscribed in the ellipse below.
I'm not quite sure where to start...first of all, how would you even enter this into a calculator to graph? On the TI-83, I only see one variable 'x' that you...
Homework Statement
I have been stuck working on this problem for the past little while and can't seem to figure it out.
A construction company needs to create a trucking route for five years to transport ore from a mine site to a smelter. The smelter is located on a major highway 10km from...
Homework Statement
I remember doing something very similar to this in pre-calc, but I don't know where to get started.
A candy box is to be made out of a piece of cardboard that measures 8 by 12 inches. Squares of equal size will be cit out of each corner, and then the ends and sides will...
Homework Statement
Minimize 2x²+2y²-2xy-9y subject
4x + 3y =,< 10 ,
y - 4x² =,< -2
x >,= 0
and y >,= 0.
I don't undersant this:
"This equation has no nonnegative root, which contradicts a nonnegativity
constraint."
and how we solve
-16x² + 2x + 17 + h2 = 0
I'm having a tough time solving this question.:frown: I'd appreciate it if someone can please help me out
Homework Statement
Homework Equations
y=x^2
Triangle QPR =3/4 of the area of the parabolic segment enclosed between QR and the parabola
The Attempt at a Solution
I...
How would you go about finding the highest point on the curve of intersection of an ellipsoid and a plane? Given: x^2 + y^2 + z^2 = k and ax + by + cz = j. I was thinking about using Lagrange Multipliers but I would always get stuck.
For a calc project, I am supposed to solve an interesting calculus word problem dealing with maximum and minimum values. The catch is that I cannot use my own book. Can anyone suggest a challenging optimization problem? So far, we've covered the problem with a person who must find the least time...
Homework Statement
Each edge of a square has length L. Prove that among all squares inscribed in the given square, the one of minimum area has edges of length \frac{1}{2}L\sqrt{2}
Homework Equations
The Attempt at a Solution
I started by drawing a square of sides L. Then labeled...
Homework Statement
We are given a graph of gallons of fuel per hour versus miles per hour and asked what speed should be used to maximize fuel efficiency and also what is the optimal speed(are these two the same thing).
Homework Equations
I understand optimization using the first...
Homework Statement
Homework Equations
a^2 + b^2 = c^2
A= (1/2)bh
The Attempt at a Solution
1)I labeled the distance traveled on the ocean as y and the distance traveled on land as x.
2)This one's kinda hard to describe: you know how the 100 yd distance and the shoreline form...
Homework Statement
You're building a walkway from the corner of one building to the corner of another building. The diagram looks like this.
The street is 100 ft wide, and 50 ft long.
The walkway will weigh 40 pounds per feet when it is parallel to the street and 30 pounds per feet when it is...
Here is what what written about this Lean-To:
A lean-to has a wooden floor, a 10 feet high open front, wooden side and back walls, and a wooden roof that tilts down at an angle of 45 degrees. What are the dimensions of the lean-to with the largest possible floor area that can be constructed...
Homework Statement
Find the area of the largest rectangle that can be inscribed in the region bounded by the graph of y = (4-x)/(2+x) and the coordinate aces in the first quadrant.
I think my only problem with this one is taking the derivative,
this is what i get y' = (-x^2 - 4x +...
Homework Statement
x^2 - 2xy + 6y^2 = 10
Find the point on the ellipse closest to the origin (0,0).Homework Equations
The Attempt at a Solution
Absolutely no one in my class can solve this. We've been to the math lab and none of the helpers there know how to solve it. I think the only person...
Homework Statement
At which points on the curve y=1 + 40x^3 - 3x^5 does the tangent line have th largest slope?
Homework Equations
derivative is 120x^2-15x^4...
The Attempt at a Solution
how should i do this? start by setting the derivative equal to zero to find critical #s, but...
Homework Statement
This is from the book Calculus Single Variable, 4th edition:
The cross-section of a tunnel is a rectangle of height h surmounted by a semicircular roof section of radius <i>r</i>. If the cross-sectional area is A, determine the dimensions of the cross section which...