Please help me with these questions from Math 151. Thank you so much.

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A box with a square base and gpen top_ must have a volume of 119164 cmg. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only :13, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of 23.] Simplify your formula as much as possible. it.) =1 1 Next, find the derivative, A'(a':). A'(m) :| l Now, calculate when the derivative equals zero, that is, when A'(:c) : 0. [Hint: multiply both sides by 232.] A'[a:) = 0 when a: = We next have to make sure that this value of 3: gives a minimum value for the surface area. Let's use the second derivative test. Find '4"(13). Av) =1 l Evaluate A "(m) at the zit-value you gave above. If 1900 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. Volume = cubic centimeters.Open-box Problem. An open-box (top open) is made from a rectangular material of dimensions a = 11 inches by b = 7 inches by cutting a square of side c at each corner and turning up the sides (see the figure). Determine the value of I that results in a box the maximum volume. Following the steps to solve the problem. Check Show Answer only after you have tried hard. (1) Express the volume V as a function of r: V (2) Determine the domain of the function V of I (in interval form): (3) Expand the function V for easier differentiation: V (4) Find the derivative of the function V: V (5) Find the critical point(s) in the domain of V: (6) The value of V at the left endpoint is (7) The value of V at the right endpoint is (8) The maximum volume is V (9) Answer the original question. The value of z that maximizes the volume is:A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y = 5 :32. What are the dimensions of such a rectangle with the greatest possible area? Width = l ' Height = l ' A rectangle is inscribed with its base on the xaxis and its upper corners on the parabola y = T :32. What are the dimensions of such a rectangle with the greatest possible area? Width = l l Height = l ' A cylinder is inscribed in a right circular cone of height 8 and radius (at the base] equal to 15. What are the dimensions of such a cylinder which has maximum volume? Radius = ' Height = ' A fence 15 feet tall runs parallel to a tall building at a distance of 3 ft from the building as shown in the diagram. LADDER 15 ft 3 ft Q We wish to find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building. [A] First, find a formula for the length of the ladder in terms of 0. (Hint: split the ladder into 2 parts. ) Type theta for 0. L(0 ) [B] Now, find the derivative, L'(0). Type theta for 0. L'(0) [C] Once you find the value of 0 that makes L'(0) = 0, substitute that into your original function to find the length of the shortest ladder. (Give your answer accurate to 5 decimal places.)A fence 5 feet tall runs parallel to a tall building at a distance of 4 feet from the building What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? feet A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 49 feet? ft2 A rancher wants to fence in an area of 500,000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use?Find the point on the line 2:3 + 6y + 2 = 0 which is closest to the point (3, 5). <: :w : let q="(0," and r="(5," be given points in the plane. we want to nd point p on positive at-axis such that sum of distances pq pr is as small possible. proceeding with this problem draw a picture solve need minimize following function :13: f . find has only one critical ntunber interval at :13="i" where value since smaller than values two endpoints conclude minimal distances. .-.. lj.- curve y="V" which closest b>