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Two triangular pens are built against a barn. Four hundred forty meters of fencing are to be used for the Barn three sides and the diagonal dividing fence (see figure). What dimensions maximize the area of the pen? The area of the pen is maximized if the side perpendicular to the barn is about meters long and the side parallel to the barn is about meters long. (Round to two decimal places as needed.)A right circular cylinder is placed inside a cone of radius R and height H so that the base of the cylinder lies on the base of the cone. a. Find the dimensions of the cylinder with maximum volume. Specifically, show that the volume of the maximum-volume cylinder is - the volume of the cone. b. Find the dimensions of the cylinder with maximum lateral surface area (area of the curved surface). a. Let the cylinder have radius r and height h, as shown in the figure. Find an expression for the volume V of the cylinder in terms of r and h. V= H (Type an exact answer, using it as needed.) h R.A rectangular flower garden with an area of 286 m is surrounded by a grass border 1 m wide on two sides and a 2 m wide on the other two sides as shown in the figure. What dimensions of the garden minimize the combined area of the garden and borders? 2 m Flower garden 1 m The vertical height of the garden that will minimize the total area is m. (Type an exact answer, using radicals as needed.)A boat on the ocean is 4 mi from the nearest point on a straight shoreline; that point is 10 mi from a restaurant on the shore. A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. Complete parts (a) and (b) below. 4 mi 10mi a. If she walks at 3 mi/hr and rows at 2 mi/hr, at which point on the shore should she land to minimize the total travel time? To minimize the total travel time, the boat should land miles from the restaurant. (Type an exact answer, using radicals as needed.)A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y = 361 - x". What are the dimensions of the rectangle with the maximum area? What is the area? In the rectangle with the maximum area, the shorter dimension is about and the longer dimension is about (Round to two decimal places as needed.)a. A rectangular pen is built with one side against a barn. If 2100 m of fencing are used for the other three sides of the pen, what dimensions maximize the area of the pen? b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an Barn area of 225 m (see figure). What are the dimensions of each pen that minimize the amount of fence that must be used? 225 225 225 225 a. To maximize the area of the pen, the sides perpendicular to the barn should be m long and the side parallel to the barn should be m long. (Type exact answers, using radicals as needed.)If the objective function is Q = x y and you know that x + y =16, write the objective function first in terms of x and then in terms of y. The objective function in terms of x is Q =Of all rectangles with a perimeter of 13, which one has the maximum area? (Give the dimensions.) The rectangle that has the maximum area has length and width (Simplify your answers.)Find positive numbers x and y satisfying the equation xy = 28 such that the sum 4x + y is as small as possible. The numbers are x = and y = (Type exact answers, using radicals as needed.)a. Arectangular pen is built with one side against a barn. "2191} m of fencing are used for the other three sides of the pen. what dimensions maximize the area of the pen? b. A rancher plans to make four identical and adjacent rectangular pens against a barn. each with an area of 225 m2 {see gure}. What are the dimensions of each pen that minimize the amount of fence that must he used? a. To maximize the area of the pen. the sides perpendicular to the barn should be D m long and the side parallel to the barn should be D m long. (Type exact answers. using radicals as needed.) A storage shed is to be built in the shape of a box with a square base. It is to have a volume of 275 cubic feet. The concrete for the base costs $3 per square foot, the material for the roof costs $8 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.Among all right circular cones with a slant height of 30, what are the dimensions (radius and height) that maximize the volume of the cone? The slant height of a cone is the distance from the outer edge of the base to the vertex. To maximize the volume of a right circular cone with a slant height of 30, the height must be and the radius of the base must be (Type exact answers, using radicals as needed.)An auditorium with a flat floor has a large screen on one wall. The lower edge of the screen is 5 ft above eye level and the upper edge of the screen is 13 ft above eye level (see figure). How far from the screen should you stand to maximize your viewing angle? 13 At 5 At You should stand oft from the screen to maximize your viewing angle. (Type an exact answer, using radicals as needed.)What point on the line y = 5x + 6 is closest to the origin? The point on the line y = 5x + 6 closest to the origin is (Type an ordered pair. Simplify your answer. Type integers or simplified fractions.)