what is the distance to the earths horizon from point p? enter your answer as a decimal in the box. round…

Explanation:

Step1: Apply Pythagorean theorem

Let the radius of the Earth be $r = 3959$ mi and the height of point $P$ above the Earth's surface be $h=10.2$ mi. The distance from point $P$ to the horizon $x$ forms a right - triangle with the radius of the Earth. The hypotenuse of the right - triangle is $r + h$ and the other two sides are $r$ and $x$. According to the Pythagorean theorem $x=\sqrt{(r + h)^2-r^2}$.

Step2: Expand the formula

First, expand $(r + h)^2$: $(r + h)^2=r^{2}+2rh+h^{2}$. Then $(r + h)^2-r^{2}=r^{2}+2rh + h^{2}-r^{2}=2rh+h^{2}$.

Step3: Substitute values

Substitute $r = 3959$ mi and $h = 10.2$ mi into $2rh+h^{2}$. $2\times3959\times10.2+10.2^{2}=2\times3959\times10.2 + 104.04$. $2\times3959\times10.2=2\times40381.8 = 80763.6$. $80763.6+104.04=80867.64$.

Step4: Calculate $x$

$x=\sqrt{80867.64}\approx284.4$ mi.

Answer:

$284.4$ mi