Question

1. A door of a lecture hall is in a parabolic shape. The door is 56 inches across at the bottom of the door and parallel to the floor and 32 inches high. Sketch and find the equation describing the shape of the door. If you are 22 inches tall, how far must you stand from the edge of the door to keep from hitting your head?

Answer 1

Enter the coordinate axes as shown in the picture. The equation of our parabola is y = px². The point (-16; 56) belongs to the parabola, therefore 56 = p*16². Hence p = 56/16². Find x in the picture. p(-16 + x)² = (56-22). So 256-32x + x² = 34 * 256/56. The roots of this quadratic equation are 16 +8$\sqrt{\frac{17}{7}}$ and 16-8$\sqrt{\frac{17}{7}}$ . The distance to the end of the parabola is 16-8$\sqrt{\frac{17}{7}}$

Parabola equation y = $\frac{7}{32}$ * x²