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

12.35 inches and i have attached a sketch of the parabola.

Step-by-step explanation:

We are told that height of the parabolic door is 32 inches and 56 inches across the bottom.

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

Thus, the vertex of parabola will be at A (0,32) and the equation of the parabola will be;

x² = -4a(y-32)

Now;

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

Thus; 28² = -4a(0-32)

4a × 32 = 28²

4a = 28²/32

Hence, the equation of the parabola will be;

x² = (-49/2)(y-32)

2x² = -49(y-32)

Putting y = 22, we have;

2x² = -49(22-32)

2x² = 49 × 10

Divide both sides by 2;

x² = 49 × 5

x = ±7√5

From the attached sketch,

BM = 28 - 7√5

BM = 12.35 inches

Thus; as my height is 22 inches, I am to stand 12.35 inches from edge

to keep from hitting my head

i think it might be a but you might want a second opinion