Optimization Problem-Solving Process Read the problem and identify the givens and goals. Visualize the problem, and label the quantities relevant to the problem. Find a formula for the quantity to be maximized or minimized. Use the conditions stated in the problem to eliminate variables; express the quantity to be optimized as a function of one variable. Find the first derivative, identify critical points, and locate local (relative) extrema. 2. The Fence Problem You have to enclose 3200 square feet of land {in a rectangle} using a fence on 3 sides that will cost $2 per foot, and a special wall on the fourth side that will cost $6 per foot. What are the dimensions of your enclosure that will minimize 1your cost? State the dimensions of your enclosure and the minimum cost in a complete sentence [with units!]l. 4. Another Box Problem A box with a square base and an open top needs to be constructed that will have a volume of 3200m3. What should be the dimensions of the box so that the surface area will be minimized? State the dimensions ofthe box and the minimum surface area in a complete sentence (with units!}. 5. The Furniture Problem You run a small furniture business. You sign a deal with a customer to deliver up to 400 chairs, the exact number to be determined by the customer later. The price will be $90 per chair up to 300 chairs, and above 300, the price will be reduced by $0.25 per chair (on the whole order} for every additional chair over 300 ordered. What are the largest and smallest revenues vour companv can make under this deal? State the largest and smallest revenues in a complete sentence [with units!l. 6. The Rectangle-Area Problem A rectangle has one side on the x-axis, one side on the yaxis, one vertex at the origin, and one on the curve 1! = 5e'x forx a 0' Find the maximum area of this rectangle. State the dimensions of the rectangle and the maximum area in a complete sentence [with units\". ?. The Snow Cone Problem A snow cone shaped cup is to hold approximately 118 cm3 of water. Find the height and radius of the snow cone cup that will require the least amount of paper. State the height and radius that will minimize the amount of paper in a complete sentence {with units!]