Question

A large doorway has the shape of a parabolic arch. The doorway is 20 feet high at the top of the arch and 20 feet wide at the base. At a height of 10 feet above the floor, find the width of the arch.

If you start the vertex of the parabola at 0,0, that means the equation should be x squared = 4py, right? And the other two points you know are -10, 20 and 10, 20, I think. So plugging in those numbers, you should get p = -5/4. However, the answer says 20 x square root of 1/2. Please help me figure out why this is... Thanks.

Answer 1

I would sketch a parabola with vertex at (0,20) opening downwards, and crossing the x-axis at (10,0) and (-10,0)

then in vertex form the equation would be
y = a(x-0)^2 + 20
= ax^2 + 20

but (10,0) lies on it, so
0 = a(100) + 20
x = -20/100 = -1/5

so we have y = (-1/5)x^2 + 20

then wen y = 10
10 = (-1/5)x^2 + 20
-10 = (1/5)x^2
50 = x^2
x = √50 = 7.07

so the width of the arch is 14.14 ft.

notice 20√(1/2) = 14.14