#### Answer

The minimum height is 1.95 miles.

#### Work Step by Step

Let $R$ be the Earth's radius. We can use the Pythagorean theorem to find $h$:
$(R+h)^2 = R^2+(125~mi)^2$
$R+h = \sqrt{R^2+(125~mi)^2}$
$h = \sqrt{R^2+(125~mi)^2}-R$
$h = \sqrt{(4.00\times 10^3~mi)^2+(125~mi)^2}-(4.00\times 10^3~mi)$
$h = 1.95~mi$
The minimum height is 1.95 miles.