We want to impose a coordinate system on the situation of the problem. So suppose that the base of the building is at (0,0) and that would place the top of the building at (0,40) and the the tree house is located at (25,40). After some time t the apple will have fallen s feet, which means that distance that the apple is above the ground is 40-s. Hence, the coordinates of the apple are (25,40-s).
Now, the coordinates of the light are at (0,40) and so to find where the shadow of the apple is we have to look at the line from the light to the apple, then extend that to where it hits the ground. In terms of the xy-plane, that would be where the line through (0,40) and (25,40-s) in intersects the x-axis. So, we need to find the equation of the line, through those two points.
Using the formula
m=(y2-y1)/(x2-x1),
that m=-s/25. And because (0,40) is on the y-axis, the equation for the line is
y=-(s/25)x+40.
So, to find the x-intercept we set y=0 and solve for x. This gives, 1000/s. Or in terms of t, x(t)=1000/s(t).
We are looking for dx/dt, so we take the derivative of sides of this equation for x to get
dx/dt=-1000s^(-2)ds/dt.
Now they tell us that s=16t^2. So, ds/dt=32t, and after substituting for s, we get
dx/dt=-1000/(16t)^2(32t)=-125/t^3.
To finish we plug 0.5 into t, and get -1000 ft/sec.
