7 questions to be answered I want the answers separately and work

For the given cost function C(x) = 36100 + 300x + x2 find (to 3 decimal places if needed): a) The cost at the production level 1550 b) The average cost at the production level 1550 c) The marginal cost at the production level 1550 d) The production level that will minimize the average cost e) The minimal average cost A fence 25 feet tall runs parallel to a tall building at a distance of 5 it from the building as shown in the diagram. A rectangular pen with a total area of 12700 square feet is to be divided into two parts as shown. Fencing must be used on all sides of the pen, as well as for the internal dividers. We wish to find the width that minimizes the amount of fencing required to do this. length LADDER 25 ft .width SA 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.) theta for 0. Let w represent the width of the field, and let P(w) be the amount of fencing required, as a L(0) = function of w. Find a formula for P(w). P(w) = [B] Now, find the derivative, L'(8). Type theta for 0. What width (to one decimal place) minimizes the total amount of fencing required? I' (8) w = [C] Once you find the value of 0 that makes L'(0) = 0, substitute that into your original function to ind the length of the shortest ladder, to the nearest foot. I(0 min ) A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 500 cubic centimeters of soup. The sides and bottom of the container will be made of styrofoam costing $ 0.02 (2 cents) per square centimeter. The top will be made of glued paper, costing $ 0.05 (5 cents) per square centimeter. Find the dimensions for the package that will minimize production cost. Helpful information: h : height of cylinder, r : radius of cylinder Volume of a cylinder: V = mr h Area of the sides: A = 2rrh Area of the top/bottom: A = mr2 To minimize the cost of the package: (round all answers to 2 decimal places as needed) Radius: cm Height: cm Minimum cost: cents A farmer finds that if she plants 80 trees per acre, each tree will yield 65 bushels of fruit. She 50- estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest? 40 30 trees 20 10 Give your answer rounded to the nearest whole number. - 10 10 20 3 48 Julia is a lifeguard and spots a drowning child 40 meters along the shore and 50 meters from the shore to the child. Julia runs along the shore for a while and then jumps into the water and swims from there directly to the child. Julia can run at a rate of 5 meters per second and swim at a rate of 0.8 meters per second. How far along the shore should Julia run before jumping into the water in order to save the child? Round your answer to three decimal places. A rancher plans to fence-in a rectangular corral adjacent to a river for a herd of cattle. The rancher has supplies for 1,300 meters of fence. Use calculus to determine the dimensions and area of the corral of maximum area, assuming all fencing supplies are used. The length of the shorter side of the optimal corral is meters long. The length of the longer side of the optimal corral is meters long. The optimal corral has an area of square meters