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2. Answer the following optimization problems systematically: a. A closed box with a square base is to have a volume of 2000 cm. The material for the top and bottom of the box costs P30 per square centimeter and the material for the sides cost P15 per square centimeter. Find the dimensions of the box so that the total cost of material is the least possible and all its dimensions do not exceed 20 cm. b. An open box is to be made from a 16 cm by 30 cm piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. How long should the sides of the squares be to obtain a box with the largest volume? c. An offshore oil well located at a point W that is 5 km from the closest point A on a straight shoreline. Oil is to be piped from W to a shore point B that is 8 km from A by piping it on a straight line underwater from W to some shore point P between A and B and then on to B via pipe along the shoreline. If the cost of laying pipe is P10,000,000/km underwater and P5,000,000/km over land, where should the point P be located to minimize the cost of laying the pipe? d. Find the radius and height of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 4 cm and height 3 cm.Find the extrema of the given functions on the interval [-1, 1). (a) f(x) = 3x2 -3x + 4 (b) f(x) = x3 -9x- + 15r -20 (c) f(x)=23-12-r-10 (d) f(x) = x -31/3