Math Circle 1
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UGA Math Circle March 07, 2013 Introduction We are going to see a cool technique to solve certain optimization problems. Are you thinking to use Calculus? No CALCULUS needed! Warm-up problems 1. Show that for x> 0, min(x + 1 x ) is 2. 2. A farmer wishes to put a fence around a rectangular field and then divide the field into three rect- angular plots by placing two fences parallel to one of the sides. If the farmer can afford only 1000 yards of fencing, what dimensions will give the maximum rectangular area? Arithmetic Mean-Geometric Mean inequality (A.M-G.M inequality) For n positive numbers, x 1 ,x 2 , ..., x n x 1 + x 2 + ... + x n n ≥ n √ x 1 .x 2 ...x n For n =2, x 1 + x 2 2 ≥ √ x 1 .x 2 Can you prove the A.M-G.M inequality for n = 2? 1
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Transcript of Math Circle 1
UGA Math CircleMarch 07, 2013
Introduction
We are going to see a cool technique to solve certain optimization problems. Are you thinking to useCalculus? No CALCULUS needed!
Warm-up problems
1. Show that for x > 0, min(x+1
x) is 2.
2. A farmer wishes to put a fence around a rectangular field and then divide the field into three rect-angular plots by placing two fences parallel to one of the sides. If the farmer can afford only 1000 yards offencing, what dimensions will give the maximum rectangular area?
Arithmetic Mean-Geometric Mean inequality (A.M-G.M inequality)
For n positive numbers, x1, x2, ..., xn
x1 + x2 + ...+ xnn
≥ n√x1.x2...xn
For n = 2,x1 + x2
2≥ √x1.x2
Can you prove the A.M-G.M inequality for n = 2?
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UGA Math CircleMarch 07, 2013
Activity Sheet
No CALCULUS needed!
Problems
3. We will investigate whether a Coca-Cola can could be designed to minimize the amount of aluminumused for the volume of soda it contains.
i) For a cylindrical can, closed at the top and bottom, with given volume V, find the ratio h/d of height todiameter that minimizes the total surface area A without using CALCULUS!
ii) Is the problem much easier if you are given V= 12 oz.By measuring the height and the diameter of the base of a Coca-Cola can, determine whether it minimizesthe surface for the volume it contains.
4. Use the Arithmetic Mean-Geometric Mean inequality to find the maximum volume of a box made froma 25 by 25 square sheet of cardboard by removing a small square from each corner and folding up the sidesto form a lidless box.
Challenge: a) Use the AM-GM inequality to discuss the maximum volume of a box formed from an nXnsquare sheet of cardboard.b) Can we use the AM-GM inequality when the sheet of cardboard is 20 by 25?
5. Determine the positive integer n, for which the value ofn
2+
18
nis the smallest.
6. Find three positive integers x, y, z such that their sum is 32 and P = xy2z is maximum.7. Find the maximum of the function (1− x)(1 + x)2 in the interval [0,1].8. If a, b, c, d are positive integers such that a + b + c + d = 2008, what is the largest possible value forab+ cd+ ad+ bc.
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UGA Math CircleMarch 07, 2013
Activity Sheet
No CALCULUS needed
More Problems
9. Use the Arithmetic Mean and Geometric Mean inequality to find the maximum area of a circular sectorwith a fixed perimeter.
10. Use Arithmetic and Geometric Mean inequality to find the maximum and minimum values for y =x
1 + x2
11. Find the minimum of f(x) =9x2(sin(x))2 + 4
x sin(x)in the range of 0 < x < π.
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UGA Math CircleMarch 07, 2013
12. Find the maximum value of ab(72− 3a− 4b), a > 0, b > 0.
13. Prove that if a, b, c are positive integers, then (a+ b)(b+ c)(c+ a) ≥ 8abc.
14. A farmer wants to fence in 60,000 square feet of land in a rectangular plot along a straight high-way. The fence he plans to use along the highway costs $2 per foot, while the fence for the other threesides costs $1 per foot. How much of each type of fence will he have to buy in order to keep expenses toa minimum? What is the minimum expense? (Use the AM-GM inequality to find a solution. No calculusneeded.)
15. (Geometric proof for A.M-G.M inequality) Given a right angled triangle, its altitude from the rightangle divide the hypotenuse into parts of lengths a and b. Interpret the comparison of the lengths of thealtitude and the median from the 90 degrees vertex of a right triangle having hypotenuse of length a+b.Compute the altitude and the median. What conclusion can you draw from your answer.
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UGA Math CircleMarch 07, 2013
Activity Sheet
Challenging Problems
16. Use the AM-GM inequality to find a point on the graph of y =1
x2that is closest to the origin.
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UGA Math CircleMarch 07, 2013
17. What is the maximum volume of a right circular cylinder that can be inscribed in a sphere of radius r.
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UGA Math CircleMarch 07, 2013
18. Find the minimum for x2 +1
x, x > 0.
19. Use Arithmetic Mean-Geometric Mean inequality to show that the maximum area of a triangularregion with a given perimeter is attained when the triangle is equilateral.
20. A rectangle is bounded by the x-axis and the semicircle y =√
25− x2. What length and widthshould the rectangle have so that its area is a maximum?
21. A rectangular box is resting on the xy−plane with one vertex at the origin. The opposite vertexlies in the plane 6x+ 4y + 3z = 24 with x, y, z ≥ 0. Find the maximum volume of such a box.
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