A doorway has the shape of a parabolic arch and is 16 feet high at the center and 8 feet wide at the base. If a rectangular box 7 feet high must fit through the doorway, what is the maximum width the box can have? Solution Let x = 0 in the middle of the doorway, and y = 0 at the ground. Then, y = 16 at the top. As the doorway is 8 feet wide, x = -4 on one side and 4 on the other side The parabola goes through (0,16), (-4, 0), and (4, 0) Then y = 16 - x^2 includes these 3 points. Then, at height 7, y = 16 - x^2, so 7 = 16 - x^2 x^2 = 9 x = ±3 The box can go from -3 to +3, so the box can be 6 feet wide. .