Question

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

y =32- ax² ;

y(28)=y(-28)=0;

32- a*28²=0,

$a=\frac{32}{28^2};$  

$a=\frac{2}{49}$ ;

$y=32-\frac{2}{49}x^2.$

Next, one should find x such that $y \geq 22$

$32-\frac{2}{49}x^2 \geq 22,$

$\frac{2}{49}x^2 \leq 10,$

$x^2 \leq \frac{490}{2},$

$-7 \sqrt{5} \leq x \leq 7 \sqrt{5},$

-15.66<x<15.66

If the vertex of parabola is (0,32), then a person should be located between -15.66 and 15.66, that is, approximately 15.66 inches width to the left of the vertex of the parabola and approximately 15.66 inches width to the right of the vertex of the parabola where the parabola is higher than the height of 22 inches.