Question 1117083
A related-rates problem.

Let 3rd base be at the origin, 2nd base at (0,90) and  H the home plate along the x-axis at (90,0).

r is the distance from the runner to home plate.<br>

Instantaneous values at t=t0:
y(t0) = 30 ft
r(t0) = {{{ sqrt(30^2 + 90^2) = sqrt(9000) = 30*sqrt(10) }}} ft 
dy/dt = -28ft/min   (negative because the direction is down the y-axis, so y values are getting smaller)<br>

The problem asks you to find <b> dr/dt. </b>

r = {{{ sqrt(x^2+y^2) }}}   where  r=r(t), x=x(t), y=y(t) 
Since x(t) is a constant (H is at (90,0), so the x distance is 90ft),  we have
r = {{{ sqrt(8100 + y^2) }}}
Differentiating both sides WRT t:
 
dr/dt = {{{ (1/2)(8100+y^2)^(-1/2)*2y*(dy/dt) =  (y/sqrt(8100+y^2))*(dy/dt) }}}

dr/dt =  {{{ 30*(1/sqrt(8100+30^2))*(-28) = (30/sqrt(9000))*(-28) = highlight( -8.85438 ) }}} ft/min