Parabola Problem

Question

A simply supported beam is 64 feet long and has a load at the center (see figure). The deflection (bending) of the beam at its center is 1 inch. The shape of the deflected beam is parabolic. https://www.webassign.net/larprecalcaga5/10-1-090.gif

(a) Find an equation of the parabola. (Assume that the origin is at the center of the beam. Express x and y in feet.)

(b) How far from the center of the beam is the deflection equal to 1/3 inch? (Round your answer to one decimal place.)

I know that the answer to a is y=(1/12288)x^2, but I have absolutely no idea why.

I have no idea how to go about solving b.

Any information is much appreciated, thanks :)

Answer 1

Since the parabola's vertex is the origin and we're talking of an upwards parabola, it looks like $\;y=ax^2\;,\;\;a>0\;$ . Now observe the parabola passes through the point $\;(32\cdot12,1)\;$ (in inches and assuming we indeed have $\;12\;$ inches in one feet)..

Find now $\;a\;$ and yoour parabola's formula.

Answer 2

$y$ should be smallest in the center. You should specify your units, as the question talks both about feet and inches. I assume the units are feet. I will take $y=0$ to be the position of the beam before the deflection. Then the beam is at $y=0$ at the ends, which are $x=\pm 32$ and $y=-\frac 1{12}$ at the center. The equation is then $y=-\frac 1{12}+\frac {x^2}{12288}$. The deflection at any point is just $|y|$, so solve $y=\frac {-1}{36}$