Answer
The well is 46.1 meters deep.
Work Step by Step
Let $y$ be the depth of the well. Let $t$ be the time it takes the stone to fall. We can write an expression for $y$:
$y = \frac{1}{2}at^2$
The time it takes for the sound to travel is $3.2-t$. We can write an expression for $y$:
$y = 343~(3.2-t) = 1097.6-343t$
We can equate the two expressions for $y$ and solve for $t$:
$\frac{1}{2}at^2 = 1097.6-343t$
$4.9t^2 +343t - 1097.6 = 0$
We can use the quadratic formula to find $t$:
$t = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$
$t = \frac{-343 \pm\sqrt{(343)^2 - 4(4.9)(-1097.6)}}{(2)(4.9)}$
$t = -73.07~s, 3.066~s$
Since $t$ must be positive, $t = 3.066~s$. We can use $t$ to find $y$, the depth of the well:
$y = \frac{1}{2}at^2$
$y = \frac{1}{2}(9.80~m/s^2)(3.066~s)^2$
$y = 46.1~m$
The well is 46.1 meters deep.
