Number of results: 24,776
Optimization
What are some ethical issues that could surface in the business world when using linear optimization techniques
Tuesday, June 15, 2010 at 11:45pm by Hickey
Optimization - Calculus
Thanks both of you.
Tuesday, December 11, 2007 at 7:41pm by Anonymous
Calculus optimization problem
85000
Sunday, October 30, 2011 at 4:40pm by sean
Calculus Optimization Problem
I agree. You are welcome.
Wednesday, March 6, 2013 at 1:10pm by Dr. Jane
Calculus (Optimization)
I mean length + 4x
Friday, December 16, 2011 at 6:23pm by Damon
Calculus Optimization
did you mean your equation to be y = -x^2/2e + e^x ??
Tuesday, January 18, 2011 at 5:05pm by Reiny
Calculus (Optimization)
Now I'm lost, I don't get why you changed the signs.
Friday, December 16, 2011 at 8:20pm by Mishaka
Calculus (Optimization)
Ok, your equation is right. Recheck your final signs as I stated.
Friday, December 16, 2011 at 8:20pm by bobpursley
Calculus (Optimization)
I dont see how you got the signs as you did. Please recheck
Friday, December 16, 2011 at 8:20pm by bobpursley
calculus
i am struggling with the concept of optimization. does anyone have any hints on how to solve these problems???
Wednesday, December 10, 2008 at 3:54pm by Hannah
Optimization - Calculus
Yes, and take the derivative and set to zero for the min distance.
Tuesday, December 11, 2007 at 7:41pm by bobpursley
Optimization - Calculus
Yes, and take the derivative and set to zero for the min distance.
Tuesday, December 11, 2007 at 7:41pm by bobpursley
Applied Calculus
I had to miss my last week of Calculus due to some personal things, so I had not been through any lectures of optimization. It hurt me greatly, now I have a few of these that I don't know how to set up at all.
Wednesday, March 13, 2013 at 5:04pm by Jacob
Calculus Optimization Problem
Thank you! I solved it out, and I got x=5 and y= 10 with a product of 500. Is this correct
Wednesday, March 6, 2013 at 1:10pm by Mary
calculus optimization
i mean find the dmensions of a drum that has a volume of 10 cubic feet and minizes the total cost
Tuesday, April 6, 2010 at 7:20pm by MILEY
Optimization - Calculus
You forgot the squares: d = [(x - 3)^2 + y^2]1/2 It is easier to minimize d^2: d^2 = (x-3)^2 + y^2 Insert in here y^2 = x+1 and set the derivative of d^2 w.r.t. x equal to zero.
Tuesday, December 11, 2007 at 7:41pm by Count Iblis
Calculus I
section is on Optimization: Find the point on the curve y = x^2 closest to the point (3, 4)
Saturday, April 14, 2012 at 11:11am by Sandra Gibson
Calculus
Find the point on the graph of y=2x-4 that is closest to the point (1,3). (Optimization equation)
Monday, December 6, 2010 at 7:28pm by Michelle
HELP!! OPTIMIZATION CALCULUS
i get a derivcative of 24w^2+24w^-1 is that correct?
Wednesday, November 23, 2011 at 5:34pm by Kay
Calculus (Optimization)
hang on, I reread the problem statement. In the first response I gave, I took your equation. I dont think it is right. give me a minute.
Friday, December 16, 2011 at 8:20pm by bobpursley
Calculus
find the derivatives, then plug it in to the orginal questioin.then use the length to find the area as it is an optimization question
Monday, January 16, 2012 at 12:27am by saphire
Calculus (Optimization)
Nevermind, that 4.42 was a mistake and my very original answer of 1.105940354 was absolutely correct!!! This is the right answer, I know it!
Friday, December 16, 2011 at 8:20pm by Mishaka
Calculus (Optimization)
Reading the question again, I think that I took the wrong interpretation and Damon took the right one.
Friday, December 16, 2011 at 6:23pm by Reiny
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...
Sunday, April 1, 2007 at 8:10am by Amy
Calculus (Optimization)
Okay, so does this change my original answer of approximately 1.64 to 4.42??? The 4.42 came from putting your new values in the quadratic equation.
Friday, December 16, 2011 at 8:20pm by Mishaka
Calculus optimization problem
Check the related question. I think you'll find that this problem has been answered many times, using different numbers.
Sunday, October 30, 2011 at 4:40pm by Steve
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)
Sunday, December 9, 2012 at 1:12am by Anonymous
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)
Sunday, October 31, 2010 at 5:22pm by Anonymous
Optimization
You might try some of the following links: http://search.yahoo.com/search?fr=mcafee&p=ethical+issues+in+the+business+world+using+linear+optimization+techniques Sra
Tuesday, June 15, 2010 at 11:45pm by SraJMcGin
calculus optimization max min
That is a problem you do when refreshed, and have some time. Here it is worked given the triangle vertexes. You are given two point, and the altidude. For your upper vertex, x,y, choose it such that the altitude (y) is 4. http://www.analyzemath.com/calculus/Problems/...
Tuesday, October 19, 2010 at 2:48pm by bobpursley
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?
Tuesday, December 11, 2007 at 7:41pm by Anonymous
Calculus - Optimization
Let L=length, then girth=84-L=2*radius=2R, or R=(84-L)/2 Volume, V = πR²L Express V in terms of L using R=(84-L)/2 Equate dV/dL=0 and solve for L.
Thursday, November 24, 2011 at 1:53am by MathMate
Calculus Optimization Problem
You need to substitute y = 15-x x(15-x)^2 x(225 -30x+x^2) 225x -30x^2 + x^3 Now you can take the derivative and set it equal to zero.
Wednesday, March 6, 2013 at 1:10pm by Dr. Jane
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.
Friday, December 16, 2011 at 6:23pm by Mishaka
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?
Thursday, May 19, 2011 at 9:10pm by K.lee
calculus test corrections. 1 question
Review your material on linear optimization. The max or min value of f(x,y) is found to be at the vertices of the region defined by the constraints. Here, the region is a triangle, with vertices at (0,0) (2,0) (0,4) f(0,0) = 1 f(2,0) = -1 f(0,4) = 5
Tuesday, April 10, 2012 at 7:48pm by Steve
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?
Tuesday, January 18, 2011 at 6:18pm by jennifer
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?
Tuesday, January 18, 2011 at 5:05pm by jennifer
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.
Wednesday, March 20, 2013 at 6:17pm by Sam
Calculus (Optimization)
Let me do some thinking... if 0=V' = 112 - 88x - 12x^2 multipy both sides by -1, and 12x^2+88x-112=0 I dont see those as your signs....
Friday, December 16, 2011 at 8:20pm by bobpursley
Calculus (Optimization)
Both of you, thank you very much!!! I arrived at the correct answer width = 18 and length = 36, but I just got that answer by chance and wasn't sure how I could prove (mathematically) that it was indeed correct, your explanations helped tremendously!
Friday, December 16, 2011 at 6:23pm by Mishaka
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?
Sunday, November 6, 2011 at 1:07am by lele
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...
Tuesday, November 16, 2010 at 9:17pm 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)
Sunday, October 31, 2010 at 5:26pm 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.)
Tuesday, March 8, 2011 at 1:01pm 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
Tuesday, October 19, 2010 at 2:48pm 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?
Tuesday, January 3, 2012 at 10:17pm by Lisa
Calculus (Optimization)
You are lost. This is algebra. if 0=112 - 88x - 12x^2 do whatever you know to put it in standard form, ax^2+bx+c=0 when you do that a and b will have the SAME signs. Surely you can do that. if a=-12, then b=-88, and c=112 if a=12, then b=88, and c=-112
Friday, December 16, 2011 at 8:20pm by bobpursley
calculus optimization problem
L + 2 W = 460 so L = (460 - 2 W) L * W = A A = (460-2W)W = 460 W - 2 W^2 DA/DW = 460 - 4 W ZERO FOR MAX OR MIN 4 W = 460 W = 115 L = 460 - 2*115 = 230
Saturday, March 30, 2013 at 1:22am by Damon
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?
Thursday, March 10, 2011 at 12:07pm by Anonymous
Calculus (Optimization, Still Need Help)
I just wanted to correct something for my equation, it should be: V = (14 - 2x)(8 - 3x)(x), which simplifies to V = 112x - 44x^2 - 4x^3. Take the derivative: V' = 112 - 88x - 12x^2 Now all I need are the roots, any help? I think I found one around 1.10594, possibly?
Friday, December 16, 2011 at 8:20pm by Mishaka
Calculus (Optimization)
I rechecked and found that 3x^2-22x+28 has the correct signs. Knowing this equation and the values I found from the quadratic equation, would you say that the 1.639079157 term is correct? (The 2.69 square inches came from squaring the 1.639079157).
Friday, December 16, 2011 at 8:20pm by Mishaka
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 ...
Wednesday, March 6, 2013 at 1:10pm by Mary
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.
Wednesday, April 3, 2013 at 3:22am by sasha
Calculus I Quick Optimization Problem
For symmetry reasons, the optimum must have a square base. Let the height of the walls be x and the side length be y Volume = x^2*y = 50 Area = x^2 +4xy = x^2 + 4x*50/x^2 = x^2 + 200/x dA/dx = 0 = 2x -200/x^2 x^3 = 100 x = 4.64 ft y = 2.32 ft
Tuesday, January 3, 2012 at 10:17pm by drwls
Calculus - Optimization
Cost=.10*(pi*r^2+2Pi*r*h)+.20PIr^2 volume= PIr^2h or h= volume/PIr^2 h=3000/(PIr^2) Put that into the cost function for h. Then take the derivative of cost with respect to r (dCost/dr), set equal to zero, solve for r.
Sunday, June 3, 2012 at 7:55pm by bobpursley
calculus optimization max min
The height of the rectangle can be anything from 0 to 4. Call it h. The width (4) of the rectangle varies linearly from 12 to 0, with w = 12 (1 - h/4)= 12 - 3h Area = f(h) = h*w = 12h - 3h^2 dA/dx = 0 when 6h = 12 h = 2; w = 12 - 6 = 6 Amax = 12 The triangle does not have to ...
Tuesday, October 19, 2010 at 2:48pm by drwls
optimization
5
Saturday, October 3, 2009 at 10:55pm by sam
Calculus (Optimization)
set V'=0, and you have a quadratic. Why not use the quadratic equation.. 12x^2+88x-112=0 3x^2+22x-28=0 x= (-22+-sqrt (22^2+12*28))/6 doing it in my head, I get about.. (-22+-28)/6= 4/6, 7.5 in my head. check my work and estimates.
Friday, December 16, 2011 at 8:20pm by bobpursley
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 ...
Monday, October 27, 2008 at 9:58am 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 ...
Saturday, October 25, 2008 at 2:40pm by George
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
Tuesday, April 6, 2010 at 7:20pm by MILEY
optimization
sry
Saturday, October 3, 2009 at 10:55pm by Anonymous
Calculus (Optimization)
I think that you might have gotten the equation wrong, I think that it should be: 3x^2 - 22x + 28. When I put this equation into the quadratic equation, I got 5.694254177 and 1.639079157. So the squares that need to be cut out should have an area of approximately 2.69 square ...
Friday, December 16, 2011 at 8:20pm 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 5ð 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?
Wednesday, March 24, 2010 at 5:06pm by Joey
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?
Saturday, March 30, 2013 at 1:22am by lori
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 ...
Saturday, October 25, 2008 at 12:37pm by George
math
This is a optimization problem !!
Friday, October 19, 2012 at 5:42pm by some one
Calculuss--Optimization
Thank you so much! I appreciate it!
Monday, November 21, 2011 at 8:47pm by Maria
Mathematics optimization
f'(c)=(b^2-a^2)/(b-a)=b+a=2c=>c=(a+b)/2
Saturday, September 3, 2011 at 7:33pm by Mgraph
Math OPTIMIZATION
hey thanks! but where did you get 16 in b?
Saturday, September 3, 2011 at 1:47am by Willoby
Basic Calculus-Optimization Problems
Since both x and y have to be positive numbers, you simply have to find for what values 200 = 4x+3y lies in the first quadrant. Find the x and y intercepts let x = 0 , y = 200/3 let y = 0 , x = 50 so 0 < x < 50 0 < y < 200/3
Friday, March 16, 2012 at 1:55am by Reiny
eco/365
what is sub optimization need an example of it to
Sunday, November 4, 2012 at 12:40pm by at
Related rates
i'm sorry, this is about optimization problems.
Sunday, May 20, 2012 at 9:02pm by :)
geometry
i almost thought this was an optimization problem
Monday, February 15, 2010 at 7:04pm by mike
HELP!! OPTIMIZATION CALCULUS
This is pretty easy. area sides=2L*m+2Wm area bottom= LW area lid= LW Volume=lwm But l=2w volume=2w^2 m or m= 4/w^2 costfunction= 4*basearea+8(toparea+sides) Now, write that cost function in terms of w (substitute) take the derivatative. with respect to w, set to zero,and ...
Wednesday, November 23, 2011 at 5:34pm by bobpursley
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...
Tuesday, December 22, 2009 at 9:20pm by Jake
Calc-optimization
Given y=(x)^1/2, find the closest point to (3/2,0)
Saturday, December 15, 2012 at 9:28pm by Daryl
Optimization (Math)
I understand now, thanks a lot steve!
Tuesday, November 1, 2011 at 12:08pm by Tommy
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 ...
Sunday, June 3, 2012 at 7:55pm by Nevin
Calculus (Optimization)
v = vol = x^2 y girth = 4 x length = y so y + 4x </=108 since maximizing y + 4x = 108 or y = 108 - 4x v = x^2 (108-4x) v = 108 x^2 - 4 x^3 dv/dx = 216 x - 12 x^2 = 0 for max or min so x(216 - 12x) = 0 x = 18 for max then y = 108 -4(18) = 36
Friday, December 16, 2011 at 6:23pm by Damon
soc
What is YOUR answer? Have you researched the definition for each of the terms here? You might try some of the following links: http://search.yahoo.com/search?fr=mcafee&p=ethical+issues+in+the+business+world+using+linear+optimization+techniques (I would have said a Civil ...
Wednesday, June 16, 2010 at 9:16am by SraJMcGin
Calculus (optimization problem)
Volume = V = pi r^2 h = constant so pi r^2 = V/h and r =(V/[pi h])^.5 and cost = 5 pi r^2 + 2 pi r h cost = 5 V/h + 2 pi r h cost = 5 V/h + 2 pi (V/pi)^.5 h^.5 d cost/dh = -5 V/h^2 + 2 pi (V/pi)^.5 (.5)(h^-.5) 0 when 5 V/h^2 = pi^.5 V^.5 h^-.5 h^1.5 = 5 V^.5/pi^.5 h^3 = 25 V/pi
Wednesday, March 24, 2010 at 5:06pm by Damon
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...
Friday, December 16, 2011 at 8:20pm by Mishaka
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 ...
Thursday, November 24, 2011 at 1:53am by Ass11
Optimization (Math)
sorry for the drawing, apparently it didn't work when I submitted the question!
Tuesday, November 1, 2011 at 12:08pm by Tommy
Calculus (Optimization)
It is just length + x^2 not girth in both directions http://pe.usps.com/text/qsg300/Q401.htm
Friday, December 16, 2011 at 6:23pm by Damon
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 (...
Thursday, November 29, 2012 at 9:50am 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...
Sunday, October 30, 2011 at 4:40pm by Sam
optimization calculus
Let length be L and width of open end be W. Height is then 144/(LW) Area of netting required A = LW (top) + WH (end) + 2LH (sides) A = LW + 144/L + 288/W Differentiate to find dA/dL and dA/dW and equate both to zero. Solve these as a pair of simultaneous equations. You should ...
Tuesday, March 8, 2011 at 1:01pm by Daniel
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 ...
Sunday, January 2, 2011 at 2:20pm by tiana
Calculus - Optimization
Question Part Points Submissions Used A parcel service will only accept packages with a length of no more than 96 inches and length plus girth of no more than 156 inches. (See the figure below.) Assuming that the front face of the package (as shown in the figure) is square, ...
Thursday, November 24, 2011 at 1:53am by Yenn
Optimization (Math)
Rats. y = (P-6P/10)/2 = 4P/20 = P/5 But then, we knew that, since y=2x
Tuesday, November 1, 2011 at 12:08pm by Steve
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 $6000/...
Wednesday, March 17, 2010 at 1:48pm by John
optimization calculus
a real estate office manages 50 apartments in downtown building . when the rent is 900$ 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 ...
Thursday, March 10, 2011 at 11:13am by Anonymous
calculus optimization
let the radius be r ft and the height h ft πr^2h = 10 h = 10/(πr^2) cost = 6πr^2 + 4πr^2 + 4(2πrh) = 10πr§2 + 8πr(10/(πr^2)) = 10πr^2 + 80/r d(cost)/dr = 20πr - 80/r^2 = 0 for max/...
Tuesday, April 6, 2010 at 7:20pm by Reiny
Math
Explain why the vertices of a solution region are important when using linear systems of inequalities for optimization problems ?
Wednesday, April 20, 2011 at 7:46pm by Tina
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 ...
Wednesday, November 23, 2011 at 5:34pm by Kay
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 ...
Wednesday, November 18, 2009 at 4:43pm by Merit
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 ...
Sunday, November 20, 2011 at 7:41pm by A
Pages: 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | Next>>
For Further Reading