Please help with numbers 3 and 6

Optimization and Related Rates Practice Problems Do the problems on your own paper. Show all work. 1) You are given a piece of cardboard that is 50 cm by 20 cm and we are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will give a maximum volume. 50 cm 20 cm _ 2) A rectangular garden is to be constructed using a rock wall as one side of the garden and wire fencing around the other three sides. Given 100 ft. of wire fencing, determine the dimensions that would create a garden of maximum area. What is the maximum area? 3) A rectangle is to be inscribed in the ellipse - + y2 = 1. What would be the dimensions of the maximum area? Find the maximum area. (A diagram is shown below.) 4) Find two positive numbers whose sum is 300 and whose product is a maximum. 5) A thin sheet of ice is in the form of a circle. If the ice is melting in such a way that the area of the sheet is decreasing at a rate of 0.5 m /sec at what rate is the radius decreasing when the area of the sheet is 12 m2? (Hint: A = mr2) 6) The angle of elevation is the angle formed by a horizontal line and a line joining the observer's eye to an object above the horizontal line. A person is 500 feet way from the launch point of a hot air balloon. The hot air balloon is starting to come back down at a rate of 15 ft/sec. At what rate is the angle of elevation, 0, changing when the hot air balloon is 200 feet above the ground. Balloon 0, Angle of Elevation Person