problems on optimization

OVERVIEW OF PROBLEMS Find two numbers whose product is 100 and whose sum is minimum. 1 Find the dimensions of a rectangle with area 1000 whose perimeter is as small as possible. 2

OVERVIEW OF PROBLEMS A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence? 3

overview of Problems If 1200 of material is available to make a box with a square base and an open top, find the largest possible volume of the box. 4 Find the point on the curve that is nearest to the point . 5

OVERVIEW OF PROBLEMS Find the points on the ellipse that are farthest away from the point . 6 Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r. 7

OVERVIEW OF PROBLEMS Consider all triangles formed by lines passing through the point and both the x- and y-axes. Find the dimensions of the triangle with the shortest hypotenuse. 8

OVERVIEW OF PROBLEMS A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible volume of such a cylinder. 9 A cylindrical can without a top is made to contain V of liquid. Find the dimensions that will minimize the cost of the metal to make the can. 10

OVERVIEW OF PROBLEMS Two poles, 6 meters tall and 15 meters tall, are 20 meters apart.  A wire is attached to the top of each pole and it is also secured to the ground somewhere between the two poles.  Where should the wire be secured so that the amount of wire used is minimum? 11

OVERVIEW OF PROBLEMS A container to hold water is formed from a metal sheet of width 80 cm by folding 20 cm from each end as shown in the figure. Find the angle so that the amount of water held by the maximum. 12

OVERVIEW OF PROBLEMS Let a and b be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point . 13