Question

the circle at the right represents a planet. the radius of the planet is about 6100 km. find the distance d to the horizon that a person can see on a clear day from the following height h above the planet.h = 7 kma person can see a distance of approximately □ km on a clear day from a height of 7 km above the given planet.(round to the nearest tenth as needed )

Explanation:

Step1: Define variables and relationship

Let $r = 6100$ km (planet radius), $h = 7$ km (height above surface). The distance $d$ forms a right triangle with $r$ and $r+h$, so use Pythagoras:
$$(r+h)^2 = r^2 + d^2$$

Step2: Rearrange for $d^2$

Isolate the term with $d$:
$$d^2 = (r+h)^2 - r^2$$

Step3: Expand and simplify

Expand $(r+h)^2$ and simplify:
$$d^2 = r^2 + 2rh + h^2 - r^2 = 2rh + h^2$$

Step4: Substitute values

Plug in $r=6100$, $h=7$:
$$d^2 = 2(6100)(7) + 7^2 = 85400 + 49 = 85449$$

Step5: Solve for $d$

Take the square root and round:
$$d = \sqrt{85449} \approx 292.3$$

Answer:

292.3 km