Hot air balloon Related Rates 
A man in a hot air balloon is ascending at a rate of $10\frac{ft}{sec}$. How fast is the distance from the balloon to the horizon ( the distance the man can see ) increasing when the balloon is $1,000$ feet high. (Hint: assume the earth is a ball of radius $4000$ miles).

My Attempt: Please tell me where it is wrong.
First we construct a triangle such that,
$x=$ radius of the earth in feet $=4000*5280$
$y=$ height of ballon in feet $=1000$
$z=$ distance the man can see in feet $=\sqrt{1000^2+(4000\times5280)^2}$
So, $$\frac{dx}{dt}=0, \frac{dy}{dt}=10\frac{ft}{sec}$$
We need to solve for $\frac{dz}{dt}$.
$$x^2+y^2=z^2 \implies 2x\frac{dx}{dt}+2y\frac{dy}{dt}=2z\frac{dz}{dt} \implies \frac{dz}{dt}=\frac{x\frac{dx}{dt}+y\frac{dy}{dt}}{z}$$
$$\frac{dz}{dt}=\frac{(4000\times5280)(0)+(1000)(10)}{\sqrt{1000^2+(4000\times5280)^2}} $$
Right here is where everything feels wrong. Is $\frac{dx}{dt}=0$? Can anyone please help me close this guy off, and did I let $x$ equal the correct value?
 A: Let $d$ be distance to horizon, $r$ be radius of earth, $h$ be the balloon's height, $s$ is balloon's speed, and $t$ is time. The OP gives $r,h,s,t$ in different units, but the units conversion in the following is implicit.
The man's line of sight is tangent to the earth at the horizon. At the horizon the line segment to the center of the earth is perpendicular to the surface of the earth and to the man's line of sight. Thus the line of sight and the line segment to the center of the earth at the horizon at the two short legs of a right triangle. The hypotenuse is the line segment from the man to the center of the earth. So $(h+r)^2=d^2+r^2 \Rightarrow \sqrt{2rh+h^2}=d$. 
$h=t*v+h_0$ giving $$d=\sqrt{2r(t*v+h_0)+(t*v+h_0)^2} \Rightarrow
d=\sqrt{v^2t^2+2v(r+h_0)t+(2r+h_0)h_0}$$
Both sides must be differentiated WRT to $t$, giving:
$$d^\prime =\frac{v^2t+v(r+h_0)}{\sqrt{v^2t^2+2v(r+h_0)t+(2r+h_0)h_0}}$$
I calculated the horizon at 1000 and 1010 feet and it matches pretty well with the formula above with $t=0$, but recommend you double check everything.
