Monday

October 20, 2014

October 20, 2014

Number of results: 13,027

**Optimization**

What are some ethical issues that could surface in the business world when using linear optimization techniques
*June 15, 2010 by Hickey*

**calculus**

i am struggling with the concept of optimization. does anyone have any hints on how to solve these problems???
*December 10, 2008 by Hannah*

**Calculus**

Find the point on the graph of y=2x-4 that is closest to the point (1,3). (Optimization equation)
*December 6, 2010 by Michelle*

**Calculus I**

section is on Optimization: Find the point on the curve y = x^2 closest to the point (3, 4)
*April 14, 2012 by Sandra Gibson*

**Calulus/Optimization**

I don't understand how to solve optimization problems, (like here's the volume of a box, find the least amount of material it would take to make such a box). Is there a tutorial or some general step by step instruction on how to do these? Thanks in advance, Amy :) http://...
*April 1, 2007 by Amy*

**AP Calculus**

A cardboard box of 108in cubed volume with a square base and no top constructed. Find the minimum area of the cardboard needed. (Optimization)
*October 31, 2010 by Anonymous*

**calculus**

optimization find the point on the graph of the function that is closest to the given point f(X)= square root of x point:(8,0)
*December 9, 2012 by Anonymous*

**Optimization - Calculus**

Find the point closest to the line sqroot(X+1) from the point (3,0). d = [(x - 3) + (y - 0)]^1/2 d = [(x - 3) + (y)]1/2 Do I now substitute in the equation y = sqroot(X+1) and solve?
*December 11, 2007 by Anonymous*

**Math - Calculus I**

Optimization Problem: Find the dimensions of the right circular cylinder of greatest volume inscribed in a right circular cone of radius 10" and height 24"
*December 5, 2013 by Alex*

**Calculus 12 Optimization**

A farmer wishes to make two rectangular enclosures with no fence along the river and a 10m opening for a tractor to enter. If 1034 m of fence is available, what will the dimension of each enclosure be for their areas to be a maximum?
*May 19, 2011 by K.lee*

**Calculus Optimization**

A model space shuttle is propelled into the air and is described by the equation y=-x^2/2e + ex (in 1000 ft), where y is its height above the ground. What is the maximum height that the shuttle reaches?
*January 18, 2011 by jennifer*

**Calculus (Optimization)**

The U.S. Post Office will accept a box for shipment only if the sum of the length and girth (distance around) is at most 108 inches. Find the dimensions of the largest acceptable box with square ends.
*December 16, 2011 by Mishaka*

**Calculus Optimization**

A model space shuttle is propelled into the air and is described by the equation y=(-x2/2e)+ex in 1000 ft, where y is its height in feet above the ground. What is the maximum height that the shuttle reaches?
*January 18, 2011 by jennifer*

**Calculus Optimization Problem**

Find two positive numbers whose sum is 15 such that the product of the first and the square of the second is maximal. I came up with this so far: x + y = 15 xy^2 is the maximum derivative of xy^2= 2xyy' + y^2 Now how do I solve this ^ after I set it to zero? I am stuck on that...
*March 6, 2013 by Mary*

**Calculus-Applied Optimization Problem: **

Find the point on the line 6x + 3y-3 =0 which is closest to the point (3,1). Note: Your answer should be a point in the xy-plane, and as such will be of the form (x-coordinate,y-coordinate)
*October 30, 2013 by Sara*

**Calculus-Applied Optimization Problem**

If a total of 1900 square centimeters of material is to be used to make a box with a square base and an open top, find the largest possible volume of such a box.
*October 31, 2013 by Ashley *

**Calculus-Applied Optimization Quiz Problem**

A rancher wants to fence in a rectangular area of 23000 square feet in a field and then divide the region in half with a fence down the middle parallel to one side. What is the smallest length of fencing that will be required to do this?
*October 31, 2013 by Riley*

**Calculus**

Optimization An offshore oil well is 2km off the coast. The refinery is 4 km down the coast. Laying a pipe in the ocean is twice as expensive as on land. What path should the pipe follow in order to minimize the cost?
*November 6, 2011 by lele*

**Calculus - Optimization **

The cost of fuel for a boat is one half the cube of the speed on knots plus 216/hour. Find the most economical speed for the boat if it goes on a 500 nautical mile trip.
*March 20, 2013 by Sam*

**Calc**

A 100 inch piece of wire is divided into 2 pieces and each piece is bent into a square. How should this be done in order of minimize the sum of the areas of the 2 squares? a) express the sum of the areas of the squares in terms of the lengths of x and y of the 2 pieces b) what...
*November 16, 2010 by katie*

**AP Calculus**

The sum of the two bases and the altitude of a trapezoid is 16ft. a) Define the area A of the trapezoid as function of its altitude. b) Find the altitude for which the trapezoid has the largest possible are. (Optimization)
*October 31, 2010 by Anonymous*

**optimization calculus**

a net enclosurefor practisinggolf shots is open at one end, as shown, find the dimensions that will minimize the amount of netting needed and give a volume of 144 m^3(netting is required only the sides, the top, the far end.)
*March 8, 2011 by Anonymous*

**calculus optimization max min**

find the dimensions of the rectangular area of maximum area which can be laid out within a triangle of base 12 and altitude 4 if one side of the rectangle lies on the base of the triangle thanks
*October 19, 2010 by Oswaldo*

**Calculus I Quick Optimization Problem**

Could you please explain this problem step by step, thank you! You are planning to make an open rectangular box that will hold a volume of 50 cubed feet. What are the dimensions of the box with minimum surface area?
*January 3, 2012 by Lisa*

**optimization calculus**

sandy is making a closed rectangular jewwellery box with a square base from two different woods . the wood for top and bottom costs $20/m^2. the wood for the side costs $30/m^2 . find dimensions that minimize cost of wood for a volume 4000cm^3?
*March 10, 2011 by Anonymous*

**calculus optimization problem**

by cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. if the cardboard is 30 inches long and 14 inches wide find the dimensions of the box that will yield the maximum volume.
*April 3, 2013 by sasha*

**calculus optimization**

a company manufactures large cylindrical drums.the bottom and sides are made from a metal that costs $4.00 a square foot, while the reinforced lid costs $6.00 a square foot. ind thedmensions ofa drm that hasa volume of 10cubic feet and minizes the total cost
*April 6, 2010 by MILEY*

**Calculus**

Explain the global optimization process for a continuous function over a closed interval. Be sure to identify all steps involved and clearly explain how the derivative is utilized in this process. Does this have to do with the first derivative rule or second derivative rule ...
*October 25, 2008 by George*

**Calculus**

Explain the global optimization process for a continuous function over a closed interval. Be sure to identify all steps involved and clearly explain how the derivative is utilized in this process. Does this have to do with the first derivative rule or second derivative rule ...
*October 27, 2008 by George*

**Calculus (Optimization)**

A rectangular piece of cardboard, 8 inches by 14 inches, is used to make an open top box by cutting out a small square from each corner and bending up the sides. What size square should be cut from each corner for the box to have the maximum volume? So far I have: V = (14 - 2x...
*December 16, 2011 by Mishaka*

**Calculus (optimization problem)**

A cyclinderical tank with no top is to be built from stainless steel with a copper bottom. The tank is to have a volume of 5š m^3. if the price of copper is five times the price of stainless steel, what should be the dimensions of the tank so that the cost is a minimum?
*March 24, 2010 by Joey*

**Calculus - Optimization**

A cylindrical container with a volume of 3000 cm^3 is constructed from two types of material. The side and bottom of the container cost $0.10/cm^2 and the top of the container costs $0.20/cm^2. a) Determine the radius and height that will minimize the cost. b) Determine the ...
*June 3, 2012 by Nevin*

**calculus optimization problem **

A farmer has 460 feet of fencing with which to enclose a rectangular grazing pen next to a barn. The farmer will use the barn as one side of the pen, and will use the fencing for the other three sides. find the dimension of the pen with the maximum area?
*March 30, 2013 by lori*

**calculus**

Optimization At 1:00 PM ship A is 30 miles due south of ship B and is sailing north at a rate of 15mph. If ship B is sailing due west at a rate of 10mph, at what time will the distance between the two ships be minimal? will the come within 18 miles of each other? The answer is...
*December 22, 2009 by Jake*

**Calculus - Optimization **

A fence is to be built to enclose a rectangular area of 800 square feet. The fence along 3 sides is to be made of material $4 per foot. The material for the fourth side costs $12 per foot. Find the dimensions of the rectangle that will allow for the most economical fence to be...
*November 17, 2013 by Jess*

**Calculus (Global Max)**

Explain the global optimization process for a continuous function over a closed interval. Be sure to identify all steps involved and clearly explain how the derivative is utilized in this process. Does this have to do with the first derivative rule or second derivative rule ...
*October 25, 2008 by George*

**calculus (optimization)**

a rectangular study area is to be enclosed by a fence and divided into two equal parts, with the fence running along the division parallel to one of the sides. if the total area is 384 square feet, find the dimensions of the study area that will minimize the total length of ...
*November 30, 2013 by yareli*

**Math**

Solve the optimization problem. Minimize F = x^2 + y^2 with x + 2y = 15. Thank You for the help!!
*June 4, 2014 by Jerry*

**eco/365**

what is sub optimization need an example of it to
*November 4, 2012 by at*

**Calc-optimization**

Given y=(x)^1/2, find the closest point to (3/2,0)
*December 15, 2012 by Daryl*

**Calculus Optimization**

An electric utility is required to run a cable from a transformer station on the shore of a lake to an island. The island is 6 km from the shore and the station is 12 km down the shoreline from a point opposite the island. It costs $4000/km to run the cable on land and $6000/...
*March 17, 2010 by John*

**Calculus - Optimization**

UBC parcel post regulations states that packages must have length plus girth of no more than 84 inches. Find the dimension of the cylindrical package of greatest volume that is mailable by parcel post. What is the greatest volume? Make a sketch to indicate your variables. I ...
*November 24, 2011 by Ass11*

**calculus**

i understand optimization but im stuck on this problem.. mrs.day is building a new deck. she has 580 square feet to enclose. if one side is bounded by a wall, find the minimum cost to build the deck if one pair of opposite sides cost $12 per foot and remaining sides cost $19 ...
*January 2, 2011 by tiana*

**Calculus**

*Optimization problem* I'm okay at some optimization problems, but this one has me stumped. You work for a company that manufactures circular cylindrical steel drums that can be used to transport various petroleum products. Your assignment is to determine the dimensions (...
*November 29, 2012 by Zared*

**Calculus optimization problem**

A rectangular dog run is to contain 864 ft ^2. a. If the dog's owner must pay for fencing, what should be the dimensions of the run to minimize cost? b. Suppose a neighbor has agreed to let the owner use an already constructed fence for one side of the run. What should the ...
*October 30, 2011 by Sam*

**optimization calculus**

a real estate office manages 50 apartments in downtown building . when the rent is 900$ per month, all the units are occupied. for every 25$ increase in rent, one unit becomes vacant. on average , all units require 75$ in maintenance and repairs each month. how much rent ...
*March 10, 2011 by Anonymous*

**calculus**

I got half of this problem wrong and I DO NOT know where and how to fix. I cannot use my calculator and have to show my work. Question: You have a 500 metre roll of fencing and a large field. You want to construct a rectangular playground area. a.) using optimization ...
*December 18, 2013 by jj*

**Calculus Optimization**

The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 500 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 10 dollar increase in rent. Similarly, one ...
*November 18, 2009 by Merit*

**Math**

Explain why the vertices of a solution region are important when using linear systems of inequalities for optimization problems ?
*April 20, 2011 by Tina*

**Calculus**

Explain the global optimization process for a continuous function over a closed interval. Be sure to identify all steps involved and clearly explain how the derivative is utilized in this process. Would this be a good explanation? The process of global optimization refers to ...
*October 27, 2008 by George*

**Calculus optimization**

A rectangular storage container with a lid is to have a volume of 8 m. The length of its base is twice the width. Material for the base costs $4 per m. Material for the sides and lid costs $8 per m. Find the dimensions of the container which will minimize cost and the minimum ...
*November 20, 2011 by A*

**HELP!! OPTIMIZATION CALCULUS **

A rectangular storage container with a lid is to have a volume of 8 m. The length of its base is twice the width. Material for the base costs $4 per m. Material for the sides and lid costs $8 per m. Find the dimensions of the container which will minimize cost and the minimum ...
*November 23, 2011 by Kay*

**Calculus - Optimization **

A parcel delivery service a package only of the length plus girth (distance around) does not exceed 24 inches. A) Find the dimensions of a rectangular box with square ends that satisfies the delivery service's restriction and has a maximum volume. What is the maximum volume? B...
*November 16, 2013 by Jess*

**a ton of calc**

basically, my teacher gave us a bunch of optimization problems and i've been working on them for hours and can't get them. if i could have help with maybe the first four, that would be AWESOME. thanks. 1) find the point on the graph of the function y = x^2 that is closest to ...
*September 9, 2009 by bleh*

**Math**

Hi I have optimization Qs with MATLAB can you help me and did you know about MATLAB cheers
*June 7, 2007 by Medo*

**optimization**

A farmer wants to make 9 identical rectangular enclosures as shown in the diagram below. If he has 720 feet of fencing materials, what should the dimensions of each enclosure be if the total area is to be maximized?
*March 8, 2014 by dillon*

**Calculus I**

*Optimization problem* [I don't know how to get to the answer] Problem: You have a cylindrical can with radius 4cm and height 10cm. Inside is a marble with a radius that has to be larger than 0 but less than 4 cm (even the largest marble will fit entirely). You're filling it ...
*November 12, 2012 by Ken*

**optimization calculus**

A piece of wire 25 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. (a) How much of the wire should go to the square to maximize the total area enclosed by both figures? (b) how much of the wire should go to the square to ...
*November 20, 2011 by A*

**optimization **

Farmer taylor wants to fence a rectangular area of 1800 square feet and divided into 3 parts by fencing parallel to the shorter side. What is the minimum amount of fencing for this job?
*April 3, 2014 by JESSICA*

**Mathematics optimization**

The arithmetic mean of two numbers a and b is the number(a+b)/2. Find the value of c in the conclusion of the mean-value theorem for f(x)=x^2 on any interval [a,b].
*September 3, 2011 by Jane*

**Calc**

How close is the semi circle y= sqr.root of 16-x^2 to the point (1, sqr.root 3)? using Optimization
*January 11, 2012 by Anonymous*

**optimization**

A model space shuttle is propelled into the air and is described by the equation y= (-x2/2e) +ex (in 1000 ft), where y is its height in feet above the ground. What is the maximum height that the shuttle reaches?
*January 13, 2011 by james*

**Calculus-Applied Optimization Problem**

The manager of a large apartment complex knows from experience that 100 units will be occupied if the rent is 425 dollars per month. A market survey suggests that, on average, one additional unit will remain vacant for each 9 dollar increase in rent. Similarly, one additional ...
*October 31, 2013 by Liz*

**calculus**

this is a optimization problem A construction company has been offered a contract for $7.8 million to construct and operate a trucking route for five years to transport ore from a mine site to a smelter. The smelter is located on a major highway, and the mine is 3 km into a ...
*July 25, 2011 by robert*

**calculus**

the second number is the reciprocal of the fist and the sm is a minimum. this is my my work and i got stuck x=first number y=second number so y=1/x then what. i know the answer is 1 and 1. also can you explain why the answer is 1 and 1 and not just one. it kind oes that to ...
*October 16, 2006 by Anonymous*

**Basic Calculus-Optimization Problems**

Hello, please help me. There is a question where I have to optimize the area of a field that is being fence in like this: [|] i.e. two rectangle fields, side by side: area= 2x by y. I get to use 200ft of fencing. From that diagram and the amount of fencing I have, I made this ...
*March 16, 2012 by mary*

**Math-Optimization**

The material for the base of a box will cost three times as much as the material for the sides and top of the box. The box must have a volume of 200 meters cubed. Find the most efficient way to built this box.
*May 27, 2008 by Megan*

**Pre-Calculus/Calculus**

I am too embarassed to ask this Calculus (really pre-calculus) question in tutoring, because I know I should know. Is the inverse of f(x)=3x-1 actually f(x)=1/3x+1? How do I find it? What if it asks the same equation replaced with f to the -1 power (x)? I think I know how the ...
*July 1, 2008 by Molly*

**Calculus/ Optimization**

A truck driver, on assignment from the owner of the truck is to drive on a 300 mile stretch of highway at a constant speed of v miles per hour. According to road signs, the minimum speed allowed is 55 miles per hour and the speed limit is 70 miles per hour. The cost of gas on ...
*March 24, 2013 by Marto*

**Math OPTIMIZATION**

A home gardener plans to enclose two rectangular gardens with fencing. The dimensions of the garden: x by 12-x, y by 12-x-y a. Find the values of x and y that maximize the total area enclosed. b. What is the maximum total area enclosed? c. How many meters of fencing are needed?
*September 3, 2011 by Willoby*

**chemistry**

I am trying to calculate geometry optimization of cyclodextrin, however, the computer always show error 2070 in gaussian. I used sem-empirical pm3 and try pm6, and tried to find any method to do this work, but every try could not work. I hope any body on Jiskha can give me ...
*May 28, 2012 by bun*

**seminar MGT**

Are “lean strategies” inconsistent with the achievement of optimization? Why or why not? This site may help you formulate your answer. http://www.isr.umd.edu/~jwh2/projects/gahagan.html If you post your ideas, we'll be glad to critique them. Optimization includes cost, ...
*June 23, 2007 by Ms Brown*

**Calculus**

What is the use of Calculus? How is it use in jobs? What jobs use Calculus? Calculus is used in engineering, economics, any physical science, and in business (e.g., actuary studies and statistics).
*May 5, 2007 by Anonymous*

**Calculus**

We're doing optimization problems and this is one that I am having trouble with: Suppose a business can sell x gadgets for p=250-0.01x dollars apiece, and it costs the business c(x)= 100+25x dollars to produce x gadgets. Determine the production level and cost per gadget ...
*May 3, 2009 by Joshua*

**Calculus**

Can anyone help me with these two optimization problems? A boat leaves a dock at 9:00 P.M. and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 10:00 P.M. How many minutes after 9:00 P.M. were the two boats...
*November 30, 2009 by Steve*

**optimization calcus**

A rectangular rose garden will be surrounded by a brick wall on three sides and by a fence on the fourth side. The area of the garden will be 1000m^2. The cost of the brick wall is $192/m. The cost of the fencing is $48/m. Find the dimensions of the garden so that the cost of ...
*March 10, 2011 by Anonymous*

**chemistry**

Using the PM3 semiempirical method in HyperChem, treat planar napthalene. First build the molecule, choose SemiEmpirical methods and PM3, and perform a Geometry Optimization. Look in the "Compute>Orbitals" menu to determine the energies of the HOMO and LUMO. What is the ...
*April 22, 2013 by muneer*

**Optimization**

min 2x+y subject to: x+y+z=1 and y^2+z^2=4 Any help would greatly be apprecaited. y^2+z^2=4 ---> put y = 2 cos(theta) and z = 2 sin(theta) x+y+z=1 ----> x = 1-2(cos(theta) + sin(theta)) 2x + y = 2 - 2 cos(theta) - 4 sin(theta) It's not difficult to find the minimum of ...
*December 7, 2006 by Dre*

**Algebra**

In an interview of 50 math majors, 12 liked calculus and geometry 18 liked calculus but not algebra 4 liked calculus, algebra and geometry 25 liked calculus 15 liked geometry 10 liked algebra but neither calculus nor geometry 2 liked geometry and algebra but not calculus. Of ...
*January 11, 2010 by Anita*

**Calculus**

1. A rectangular box is to be constructed from 2 different materials. The box will have a square base and open top. The material for the bottom costs $4.25/m2. The material for the sides costs $2.5/m2. Find the dimensions of the box with the largest volume if the budget is $...
*April 3, 2007 by A24*

**Calc-optimization**

Ann wants to build an enclosed area behind her house. One wall of the enclosed area will be the back of the house. She needs the total to be 120 sq feet. She wants to minimize the cost of fence materials. For the sides (W) fence materials cost $3/ft, for the length (L) they ...
*December 18, 2012 by Daryl*

**Calculus**

What is the simplest solution to the Brachistochrone problem and the Tautochrone problem involving calculus? (I know that the cycloid is the solution but I need a simple calculus proof as to why this is the case)
*May 27, 2010 by Sam*

**math**

8. In an interview of 50 math majors, 12 liked calculus and geometry 18 liked calculus but not algebra 4 liked calculus, algebra, and geometry 25 liked calculus 15 liked geometry 10 liked algebra but neither calculus nor geometry 2 liked geometry and algebra but not calculus. ...
*November 30, 2009 by poo*

**Economics**

2.The owner-manager of Good Guys Enterprises obtains utility from income(profit) and from having the firm behave in a socially conscious manner, such as making charitable contributions or civic expenditures. Can you set up the problem and derive the optimization conditions if ...
*May 5, 2010 by Jorie*

**home economics**

The owner-manager of Good Guys enterprises obtains utility from income (profit) and from having the firm behave in a socially conscious manner, such as making charitable contributions or civic expenditures. Can you set up a problem and derive the optimization conditions if the...
*July 30, 2010 by Anonymous*

**managerial economics**

The Owner-manager of Good Guys Enterprises obtains utility from income (profit) and from having the firm behave in a socially conscious manner, such as making charitable contributions or civic expenditures. Can you set up the problem and derive the optimization conditions if ...
*August 8, 2010 by tina*

**optimization**

The manager of a large apartment complex knows from experience that 110 units will be occupied if the rent is 322 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 7 dollar increase in rent. Similarly, one ...
*April 20, 2011 by Jane*

**Calculuss--Optimization**

At t=0, ship A is 12 miles due north of ship B. Ship A travels 12 miles/hour due south, while ship B travels 8 miles/hour due east. a. Write a function for the distance between the two ships. b. At what time are the two ships closest?
*November 21, 2011 by Maria*

**Optimization--PLEASE HELP!!!**

At t=0, ship A is 12 miles due north of ship B. Ship A travels 12 miles/hour due south, while ship B travels 8 miles/hour due east. a. Write a function for the distance between the two ships. b. At what time are the two ships closest?
*November 21, 2011 by Maria*

**math - Calc optimization**

You are an engineer in charge of designing the dimensions of a box-like building. The base is rectangular in shape with width being twice as large as length. (Therefore so is the ceiling.) The volume is to be 1944000 m3. Local bylaws stipulate that the building must be no ...
*March 29, 2011 by J Velji*

**Math**

Optimization Problem A right circular cylindrical can of volume 128tπ cm^3 is to be manufactured by a company to store their newest kind of soup. They want to minimize the surface area of the can to keep costs down. What are the dimensions of the can with minimum surface ...
*March 15, 2013 by Kevin*

**math-calculus**

1.For any function y+f(x), express dy/dx using limit notation. Give a clear explanation for your given answer. you probably meant to type y=f(x) By definition dy/dx = Limit (f(x+∆x) - f(x))/∆x as ∆x-->0 The explanation of this equation is the whole basis ...
*March 26, 2007 by Chris*

**calculus**

Can someone please tell me what the word "WHOLE" AND "IRRATIONAL" MEAN IN CALCULUS? THANKS
*September 2, 2008 by Mike*

**Math:Calculus**

Use the Fundamental Theorem of Calculus to find the derivative of g(x)=1/(1+t^5)dt in the interval 1 to x.
*November 22, 2010 by HELP!!!!*

**Calculus**

Use fundamental theorem of calculus: Int(pi/2_pi) of e^(sin(q))*cos(q)dq
*April 28, 2011 by Rachel *

**Calculus Help **

Use calculus to find the area A of the triangle with the given vertices. (0, 0), (3, 1), (1, 8) A = ????
*May 1, 2014 by Anonymous*

**Calculus, bobpursley! **

This has to do with my last Calculus problem. I need help on it..The part I replied to after you answered the question.
*September 21, 2010 by Leanna*

**Calculus**

The area is bounded by y= x^3, x= 2 is rotated around the x-axis. Find the volume using calculus.
*July 24, 2012 by Ted*

**math, calculus 2 **

Use part 1 of the Fundamental Theorem of Calculus to find the derivative of g(x)= integrate from 9x to 4x of ((u+4)/(u-1))du
*August 29, 2012 by bobby*

**Calculus Precalculus**

Answer the calculus question about Point P, 45 mph, etc. It is ALREADY posted on here.
*August 27, 2011 by Zach*

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