Question 1160419
<pre>
This is a related-rates problem.

The distance h from the runner to 2nd base is {{{ h = sqrt(90^2 + x^2) }}}
where x is the remaining distance from the runner to 1st base.

Given:
{{{dx/dt = 22 (ft/s) }}} but note that because the way I've defined x, it is decreasing.  Therefore I will write {{{ dx/dt = -22 (ft/s) }}} so if we get a negative answer it indicates a decreasing distance. 

[ If you instead insist on dx/dt > 0, you can say dx/dt=22 ft/s and define x as the distance from home plate to the runner, then the distance remaining to first base is 90-x and you end up with the minus sign later, but the math is a little uglier. ]

Using the chain rule:
{{{ dh/dt = (dx/dt)(dh/dx) = -22 * (1/2)*(90^2+x^2)^(-1/2) *2x }}} = {{{ (-22x) / sqrt(90^2+x^2) }}}

Plugging in x=45 gives  {{{ highlight( dh/dt = -9.839 ) }}} ft/s
 (The runner's distance with respect to 2nd base is decreasing at 9.839 ft/s)