Solve 1,2,3,4, and Brain Teaser

1.The U.S. Post Office will accept a box for shipment only if the sum of the length and girth (distance around) is at most I03 inches. Find the dimensions of the largest acceptable box with square ends. 2. A rectangular piece of cardboard, 8 inches by 14 inches, is used to make an open top box by cutting out a small square from each comer and bending up the sides. l.l'lihat size square should be cut from each corner for the box to have the maximum volume? I-' l4\"|l i: l 3. Find the point on the graph 01'}: = x2 + l that's closest to the point 8,1.5 . Hint: Remember the distance formula. The distance between two points in a plane is d = 110:1 x2 2 + (yl y2 2. 4. Given a right circular cone, you put an upside-down cone inside it so that its vertex is at the center of the base of the larger cone, and its base is parallel to the base of the larger cone. If you choose the upside-down cone to have the largest possible volume, what fraction of the volume of the larger cone does it occupy? (Let Hand R be the height and radius of the large cone, let 1: and r be the height and radius of the small cone. Use similar triangles to get an equation relating h and r. The formula for the volume of a cone is V: %1rr2h.) Brain Teaser You're sunbathing on the island shown on the map below. The island is six miles from shore at the closest point, and the nearest store is a convenience store seven miles down the beach. If you can row at 4 miles an hour, and you can walk at 5 miles an hour, where should you land to get a bag of tortilla chips in the least possible time? (Ignore tides, currents, and sharks. No fair renting a helicopter.)