4 questions to be answered I need the answers separately and the work

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 $ 0.03 (3 cents) per square centimeter. The top will be made of glued paper, costing $ 0.08 (8 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 = Trzh 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: I cm Height cm Minimum cost: cents A box with a square base and open top must have a volume of 87808 cm . 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 , the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of x.] Simplify your formula as much as possible. A(z) = Next, find the derivative, A' (x). A' (x) = Now, calculate when the derivative equals zero, that is, when A'(x) = 0. [Hint: multiply both sides by a2. ] A'(x) = 0 when x = We next have to make sure that this value of a gives a minimum value for the surface area. Let's use the second derivative test. Find A"(I). A"(I ) = Evaluate A"(x) at the x-value you gave above. A small house was built on an island off a perfectly straight Shoreline. The point B on the shoreline that is closest to the island is exactly 7 miles from the island. Eight miles east of B is the closest source of fresh water to the island. A pipeline is to be built from the island to the source of fresh water by laying pipe underwater in a straight line from the island to a point Q on the shoreline between B and the water source, and then laying pipe on land along the shoreline from Q to the source. It costs 7000 dollars per mile to lay the pipe under the water and 4375 dollars per mile to lay pipe on land. How far east of B should Q be located in order to minimize the total construction costs? Evaluate the integral (go*+ 18x3 + 1423) da underwater underground Lol Find a formula for the cost of laying the pipe in terms of a C(z) = Find the value for a that will minimize the total cost