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

The height of the parabolic door is 32 inches. It is 56 inches across the bottom.

Let origin of the cartesian co-ordinate system is taken at the mid point of the bottom, x-axis along the bottom of the door and y-axis perpendicular to bottom through mid point of bottom.

Then vertex of parabola is at A (0,32) and so equation of the parabola will be

x$^2$ = -4a(y-32)

B(28,0) and C(-28,0) are two points on the parabola.

So 28$^2$ = -4a(0-32)

=> 4a*32 = 28*28

=> 4a = $\frac{28*28}{32} = \frac{49}{2}$

Therefore the equation of the parabola is x² = - $\frac{49}{2}$ (y-32)

=> 2x$^2$ = -49(y-32)

Putting y = 22 we get

2x$^2$ = - 49(22-32) = 49*10

=> x$^2$ = 49*5

=> x = ±$\sqrt{49*5} = ±7\sqrt{5}$

From given sketch, BM = 28-7$\sqrt{5}$

As my height is 22 inches, I am to stand (28-7$\sqrt{5}$ ) inches away from edge to keep from hitting my head

$\mathbf{Answer}$

28-7$\sqrt{5}$ inches from edge

Approximately 12.35 inches ( correct upto two places of decimal)