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### Section 4-8 : Optimization

- Find two positive numbers whose sum of six times one of them and the second is 250 and whose product is a maximum.
- Find two positive numbers whose sum of twice the first and seven times the second is 600 and whose product is a maximum.
- Let \(x\) and \(y\) be two positive numbers whose sum is 175 and \(\left( {x + 3} \right)\left( {y + 4} \right)\) is a maximum. Determine \(x\) and \(y\).
- Find two positive numbers such that the sum of the one and the square of the other is 200 and whose product is a maximum.
- Find two positive numbers whose product is 400 and such that the sum of twice the first and three times the second is a minimum.
- Find two positive numbers whose product is 250 and such that the sum of the first and four times the second is a minimum.
- Let \(x\) and \(y\) be two positive numbers such that \(y\left( {x + 2} \right) = 100\) and whose sum is a minimum. Determine \(x\) and \(y\).
- Find a positive number such that the sum of the number and its reciprocal is a minimum.
- We are going to fence in a rectangular field and have 200 feet of material to construct the fence. Determine the dimensions of the field that will enclose the maximum area.
- We are going to fence in a rectangular field. Starting at the bottom of the field and moving around the field in a counter clockwise manner the cost of material for each side is $6/ft, $9/ft, $12/ft and $14/ft respectively. If we have $1000 to buy fencing material determine the dimensions of the field that will maximize the enclosed area.
- We are going to fence in a rectangular field that encloses 75 ft
^{2}. Determine the dimensions of the field that will require the least amount of fencing material to be used. - We are going to fence in a rectangular field that encloses 200 m
^{2}. If the cost of the material for of one pair of parallel sides is $3/ft and cost of the material for the other pair of parallel sides is $8/ft determine the dimensions of the field that will minimize the cost to build the fence around the field. - Show that a rectangle with a fixed area and minimum perimeter is a square.
- Show that a rectangle with a fixed perimeter and a maximum area is a square.
- We have 350 m
^{2}of material to build a box whose base width is four times the base length. Determine the dimensions of the box that will maximize the enclosed volume. - We have $1000 to buy the materials to build a box whose base length is seven times the base width and has no top. If the material for the sides cost $10/cm
^{2}and the material for the bottom cost $15/cm^{2}determine the dimensions of the box that will maximize the enclosed volume. - We want to build a box whose base length is twice the base width and the box will enclose 80 ft
^{3}. The cost of the material of the sides is $0.5/ft^{2}and the cost of the top/bottom is $3/ft^{2}. Determine the dimensions of the box that will minimize the cost. - We want to build a box whose base is a square, has no top and will enclose 100 m
^{3}. Determine the dimensions of the box that will minimize the amount of material needed to construct the box. - We want to construct a cylindrical can with a bottom but no top that will have a volume of 65 in
^{3}. Determine the dimensions of the can that will minimize the amount of material needed to construct the can. - We want to construct a cylindrical can whose volume is 105 mm
^{3}. The material for the wall of the can costs $3/mm^{2}, the material for the bottom of the can costs $7/mm^{2}and the material for the top of the can costs $2/mm^{2}. Determine the dimensions of the can that will minimize the cost of the materials needed to construct the can. - We have a piece of cardboard that is 30 cm by 16 cm and we are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will give a maximum volume.
- We have a piece of cardboard that is 5 in by 20 in and we are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will give a maximum volume.
- A printer needs to make a poster that will have a total of 500 cm
^{2}that will have 3 cm margins on the sides and 2 cm margins on the top and bottom. What dimensions of the poster will give the largest printed area? - A printer needs to make a poster that will have a total of 125 in
^{2}that will have ½ inch margin on the bottom, 1 inch margin on the right, 2 inch margin on the left and 4 inch margin on the top. What dimensions of the poster will give the largest printed area?