Example 3: A standard soda-can holds 355 milliliters of liquid. A milliliter occupies 1 cubic centimeter of space. Suppose we wanted to construct such a can such that the can uses the least amount of aluminum. Find the dimensions of the can that does this and the amount of aluminum used. W: A box with no top is to be made from a 20 x 16 inch sheet of cardboard by cutting out one- inch squares from each comer and then folding up and gluing the remaining edges to form the sides of the box. (a) Make a sketch of this situation. (b) Find the volume of such a box. (c) If the dimension of the cutout square were 2 inches, what would be the volume of the box? ((1) m: there can be many different sized squares that may be cut out to form this box. We are interested in the size of the square which will yield the largest volume. Use calculus to nd the dimension of the cut-out square which yields the largest volume of the box. Give the dimensions of the box and the maximum volume. Example 2: A man decides to jog to the local donut shop for breakfast. He starts at his home, point A, and jogs directly to the east to point 0. Along this path, there is a trail made for running. From point 0, he then jogs directly north to get to the donut shop along a big eld. If the donut shop is located 2 miles east and 1 mile north of the man's home, then answer each of the following questions: (a) The man jogs at 10 milesfhour on the road from point A to point 0 (on the trail) but only 6 miles per hour from point 0 to the donut shop (in the eld). Find the time it would take to the man to reach the donut shop if he takes the route given above. (b) Suppose that the man instead decides to jog directly from point A to the donut shop (in a straight line) through the eld. How long would it take him to do this? (c) Suppose that the man does this every Saturday. After doing this for a couple of months, he starts to realize that while he is jogging, he just craves the donuts. He gets an idea that perhaps there is a shortest route to take. He decides that he will start trying different paths. He will jog along the trail to some point before 0, then cut across the eld to the donut shop. At what point should the man enter the field in order to reach the donut shop in minimum time? Examplej: A movie screen on a wall is 20 feet high and 10 feet above the floor. At what distance x from the front of the room should you position yourself so that the viewing angle of the movie screen is as large as possible