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PRECALCULUS GT/HONORS WORKSHEET 2 ON OPTIMIZATION In each of the following, write a function in one variable, and then use your calculator to find the answer. Draw a sketch of the function you used, label your answer on your sketch, and then write your answer in a sentence. 1. Find two positive numbers such that the sum of the first and twice the second is 100 and the product is a maximum. 2. Find two positive numbers such that the product is 48 and the sum of the first plus three times the second is a minimum. 3. A rancher has 180 feet of fencing with which to enclose four adjacent rectangular corrals as shown. What dimensions should be used so that the enclosed area will be a maximum? What will the area be? 4. A gardener wants to make a rectangular enclosure using a wall as one side and 120 meters of fencing for the other three sides. Find the dimensions of the enclosure that will give the greatest area. What will the area be? wall 5. A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. 6. A tank with a rectangular base and rectangular sides is open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. If building the tank costs $10 per square meter for the base and $5 per square meter for the sides, what is the cost of the least expensive tank? What are its dimensions? 7. A rectangle is bounded by the x-axis and the semicircle y 25 x 2 as shown. What length and width should the rectangle have so that its area is a maximum? 8. A rectangular page is to contain 24 square inches of print. The margins at the top and 1 bottom of the page are to be 1 inches, and the margins on the left and right are to 2 be 1 inch. What should the dimensions of the page be so that the least amount of paper is used? 9. Sixteen feet of wire is to be used to form a square and a circle. Find the length of the sides of the square and the radius of the circle to enclose (a) the minimum total area. (b) the maximum total area. 10. A cylindrical metal container, open at the top, is to have a capacity of 24 in 3 . The cost of material used for the bottom of the container is $0.15 per sq. in., and the cost of the material used for the curved part is $0.05 per sq. in. Find the dimensions that will minimize the cost of the material, and find the minimum cost. 11. Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. How far from the left post should the stake be placed to use the least wire? 12 ft 28 ft 30 ft