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### Section 4.8 : Optimization

1. Find two positive numbers whose sum is 300 and whose product is a maximum. Solution
2. Find two positive numbers whose product is 750 and for which the sum of one and 10 times the other is a minimum. Solution
3. Let $$x$$ and $$y$$ be two positive numbers such that $$x + 2y = 50$$ and $$\left( {x + 1} \right)\left( {y + 2} \right)$$ is a maximum. Solution
4. We are going to fence in a rectangular field. If we look at the field from above the cost of the vertical sides are $10/ft, the cost of the bottom is$2/ft and the cost of the top is $7/ft. If we have$700 determine the dimensions of the field that will maximize the enclosed area. Solution
5. We have 45 m2 of material to build a box with a square base and no top. Determine the dimensions of the box that will maximize the enclosed volume. Solution
6. We want to build a box whose base length is 6 times the base width and the box will enclose 20 in3. The cost of the material of the sides is $3/in2 and the cost of the top and bottom is$15/in2. Determine the dimensions of the box that will minimize the cost. Solution
7. We want to construct a cylindrical can with a bottom but no top that will have a volume of 30 cm3. Determine the dimensions of the can that will minimize the amount of material needed to construct the can. Solution
8. We have a piece of cardboard that is 50 cm by 20 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.
Solution