A rectangle is constructed with its base on the xaxis and its upper two vertices on the parabola y = 36 - x2. What are the dimensions of the rectangle with the maximum area? What is the area? The shorter dimension of the rectangle is and the longer dimension is (Type an exact answer, using radicals as needed.) You are designing a rectangular poster to contain 36 in.2 of printing with a 4-in. margin at the top and bottom and a 1-in. margin at each side. What overall dimensions will minimize the amount of paper used? Two sides of a triangle are 6 and 6. Find the size of the angle 9 (in radians) formed by the sides that will maximize the area 1 of the triangle. (Hint: A: [3] sh sin 9.) Find a positive number for which the sum of it and its reciprocal is the smallest (least) possible. Let x be the number and let S be the sum of the number and its reciprocal. Write the objective function in terms of x. 8(x) = A piece of wire of length 70 is cut into two pieces. One piece is bent into a square and the other is bent into a circle. If the sum of the areas enclosed by each part is a minimum, what is the length of each part? To minimize the combined area, the wire should be cut so that a length of is used for the circle and a length of is used for the square. (Round to the nearest thousandth as needed.) Suppose that C(X) = 6x3 - 24x2 + 8,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items. The production level that minimizes the average cost of making x items is x = (Simplify your answer.) A storage shed is to be built in the shape of a box with a square base. It is to have a volume of 175 cubic feet. The concrete for the base costs $5 per square foot, the material for the roof costs $2 per square foot. and the material for the sides costs $2.50 per square foot. Find the dimensions of the most economical shed. The length of one side of the shed's base is ft. The height of the shed is ft. A rectangular tank that is 4000 ft3 with a square base and open top is to be constructed of sheet steel of a given thickness. Find the dimensions of the tank with minimum weight. The dimensions of the tank with minimum weight are ft. (Simplify your answer. Use a comma to separate answers.) Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 15. 15 15 Y a: V = cubic units (Type an exact answer.) What are the dimensions of the lightest open-top right circular cylindrical can that will hold a volume of 1,331 cm3? The radius of the can is cm and its height is cm. (Type exact answers, using 1: as needed.)